Paul knows the meaning of Jesus’ saving crucifixion. He is very clear about him, the one who saves all who will. Writing to the saints (believers) in Rome he puts it like this:

Much more surely then, now that we have been justified by his blood, will we be saved through him from the wrath of God. [e] For if while we were enemies, we were reconciled to God through the death of his Son, much more surely, having been reconciled, will we be saved by his life. Romans 5:9-10 (NRSV)

“Will we be saved”. Paul looks to the future – to his, and all believers’, ultimate destiny. The “groundwork” was finished in that once and for all, full, perfect and sufficient sacrifice of Christ; the ultimate effect was and is assured. God empowers your grateful and humble acceptance of his mercy. Nothing left to do. No need for enmity with God. No “maybe” in this. It is certain, twice over! (That NRSV marginal note draws attention to the ”of God” being implied. Given his earlier discussion, can there be any doubt on the wrath Paul spoke about?)

Here is the same passage from the New Living Translation: And since we have been made right in God’s sight by the blood of Christ, he will certainly save us from God’s condemnation. For since our friendship with God was restored by the death of his Son while we were still his enemies, we will certainly be saved through the life of his Son. Romans 5 (NLT)

Clearly the Saviour will completely deliver former enemies from coming under wrath. Jesus saves! It is not a question of what is deserved but of the greatness of the salvation Jesus achieved. (I did a previous post on “salvation”.)

The word (verb) we know as “save” is frequent in the NT. (It is also translated in other ways, as shown below.) In this post I am looking at some examples:

Some familiar words from Jesus are relevant: The seeds that fell on the footpath represent those who hear the message, only to have the devil come and take it away from their hearts and prevent them from believing and being saved. Luke 8:12 (NLT). Jesus’ illustration claims hearing his message can lead to being saved, if the message is accepted, believed and retained.

In comparable vein, here are some challenging words from Jesus. If you want to save your life,[e] you will destroy it. But if you give up your life for me and for the good news, you will save it. Mark 8:35 (CEV). He speaks of retaining life intact in a different sense than simply breathing! So, Jesus wants to see people saved. Not that I accept such human testimony, but I say these things so that you may be saved. John 5:34 (NRSV)

Contrary to everyday experience, to have life in God’s kingdom is not dependent on wealth, birth or status. In fact, it is easier for a camel to go through the eye of a needle than for a rich person to enter the Kingdom of God!” The disciples were astounded. “Then who in the world can be saved?” they asked, Mark 10:25-26 (NLT).

Everything depended on Jesus and him completing his work, in face of its horror: ”Now my soul is deeply troubled. Should I pray, ‘Father, save me from this hour’? But this is the very reason I came! John 12:27 (NLT) Jesus wanted to save, not to be saved from the consequences earned by the rest of us.

Note also: Jesus calls for faithfulness in the face of animosity, in view of the final end – as he was, so may his disciples be. And you will be hated by all for my name's sake. But the one who endures to the end will be saved. Mark 13:13 (ESV).

Saving can also refer to deliverance from terrestrial destruction: And if the Lord had not cut short those days, no one would be saved; but for the sake of the elect, whom he chose, he has cut short those days. Mark 13:20 (NRSV)

In describing Jesus’ work it may be a matter of judgement or context as to use “save” or “heal”. Perhaps there is a deliberate significance that the “save” word is used in clear instances of healing or curing, etc. (Other words were available; the "save" word was not inevitable.) It could even be reasonable to consistently translate as “save”, instead of “heal”, etc.

The saver of others: On the day of Jesus’ crucifixion we find some telling words from opponents, who spoke truer than they knew. What did they mean by this scornful mockery, that Jesus only claimed he saved others?
Save yourself and come down from the cross!”
The chief priests and the teachers of the Law of Moses also made fun of Jesus. They said to each other, “He saved others, but he can’t save himself. Mark 15:30-31 (CEV). Evidently Jesus’ image was of the “healer/saver” and the “saver” role was a sticking point (as was forgiver of sins). The suggestion of "lostness" was no more acceptable than "blindness", nor "slavery". Moreover, there is an unconscious irony in those words, because Jesus accepted the cross as the path to salvation availability for humankind. (The critics can hardly be talking about healing.) In these bitter words that verb “save” can scarcely have meant “heal”.

There is an overlap of word usage for the verb, with "save" as the basic emphasis. Consider these segments which all include the same verb; some unmistakably refer to or imply healing work, with visible results:
And Jesus said to them, “I ask you, is it lawful on the Sabbath to do good or to do harm, to save life or to destroy it?” Luke 6:9 (NRSV)
And those who had seen it told them how the demon-possessed[d] man had been healed. Luke 8:36 (NRSV)  
When a sinful woman in that town found out that Jesus was there, she bought an expensive bottle of perfume. Then she came and stood behind Jesus. She cried and started washing his feet with her tears and drying them with her hair. The woman kissed his feet and poured the perfume on them. …… (text omitted). But Jesus told the woman, “Because of your faith, you are now saved.[e] May God give you peace!” Luke 7: 37,38&50 (CEV). Translators here give “healed” as the alternative to “saved”.
A desperate leader came: He knelt at Jesus' feet and started begging him for help. He said, “My daughter is about to die! Please come and touch her, so she will get well and live.” Mark 5:22-23 (CEV). It was a testing moment for this father: But Jesus on hearing this answered him, “Do not fear; only believe, and she will be well.” Luke 8:50 (ESV)
And he said to her, “Daughter, your faith has made you well. Go in peace. Your suffering is over.Mark 5:34 (NLT)
And wherever he came, in villages, cities, or countryside, they laid the sick in the marketplaces and implored him that they might touch even the fringe of his garment. And as many as touched it were made well. Mark 6:56 (ESV)

John tells us a man named Lazarus became ill and died. The account includes this misunderstanding: After saying this, he told them, “Our friend Lazarus has fallen asleep, but I am going there to awaken him.” The disciples said to him, “Lord, if he has fallen asleep, he will be all right. John 11:11-12 (NRSV). I include this because the disciples' words may be translated as ”he will be saved” (so 1881 RV margin; Oxford edition). All of the English versions I have checked translate John 11:12 in terms of recovery, rather than of being saved.

Following are two sentences from the one Acts location using the "save" word in the different ways. This illustrates the linkage to the saving work of Christ intended for all, regardless of health status: are we being questioned today because we’ve done a good deed for a crippled man? Do you want to know how he was healed? Acts 4:9 (NLT) ..... (text omitted) And there is salvation in no one else, for there is no other name under heaven given among men[c] by which we must be saved. Acts 4:12 (ESV) (The same location includes an example – in verse 10 – of an unrelated word for “healing” being used.)

Life – life eternal – is what Jesus came to bring to light and make possible for the undeserving. It is absolutely possible to be freely made right with God. The free gift does not come as humankind would guess or speculate. It bears no connection to status or importance. Nor is it a light matter, or a frivolous optional “extra”. It is no “automatic” state conferred by anyone else. Salvation by achievement is not possible. It all hinges on Jesus and relationship to him. Look at these instances on the saving:
Someone asked him, “Lord, are only a few people going to be saved?” Luke 13:23 (CEV)
Those who heard it said, “Then who can be saved?” Luke 18:26 (ESV)
• If anyone hears my words and does not keep them, I do not judge him; for I did not come to judge the world but to save the world. John 12:47 (ESV)
I am the gate. All who come in through me will be saved. Through me they will come and go and find pasture. John 10:9 (CEV)
For those who want to save their life will lose it, and those who lose their life for my sake will save it. Luke 9:24 (NRSV)

Jesus spoke bluntly of "lostness": Or those eighteen who were killed when the tower of Siloam fell on them—do you think that they were worse offenders than all the others living in Jerusalem? No, I tell you; but unless you repent, you will all perish just as they did.” Luke 13: 4-5 (NRSV). Here "lost" is translated as "perish". (The word is frequent in Luke.) It is much more than "wandered" or "misplaced". Whilst the threat of natural disaster or of Rome applied there in Jesus' time, the inward meaning and relevance comes through in these words: Then Jesus said to him, “Today salvation has come to this house, because he too is a son of Abraham. For the Son of Man came to seek out and to save the lost.” Luke 19: 9-10 (NRSV) No such thing as lost without hope!

Life-saving news was carried to the world; it is the news for today (see also Addendum below). No need to be lost. There truly is a Saviour to save you.

And he testified with many other arguments and exhorted them, saying, “Save yourselves from this corrupt generation.” Acts 2:40 (NRSV)

The free availability to all of the astounding, gracious gift of salvation by faith was hard for some to grasp. We get a glimpse at their difficulty and note the firm answer: Then certain individuals came down from Judea and were teaching the brothers, “Unless you are circumcised according to the custom of Moses, you cannot be saved. …. (text omitted) On the contrary, we believe that we will be saved through the grace of the Lord Jesus, just as they will. Acts 15:1&11 (NRSV)

Welcomed by many the gracious invitation was able to continue and it is still there. There is no distinction: the grace of the Lord Jesus is available to save all. This is not simply wishful thinking. As well as the foregoing, consider these snippets (one or two are challenging and provoke thought):
while praising God. Everyone liked them, and each day the Lord added to their group others who were being saved. Acts 2:47 (CEV) 
We were given this hope when we were saved. (If we already have something, we don’t need to hope[k] for it. Romans 8:24 (NLT) 
So you will be saved, if you honestly say, “Jesus is Lord,” and if you believe with all your heart that God raised him from death. Romans 10:9 (CEV) 
who saved us and called us with a holy calling, not according to our works but according to his own purpose and grace. This grace was given to us in Christ Jesus before the ages began, 1 Timothy 1:9 (NRSV) 
for I want somehow to make the people of Israel jealous of what you Gentiles have, so I might save some of them. Romans 11:14 (NLT) 
If you turn sinners from the wrong way, you will save them from death, and many of their sins will be forgiven. James 5:20 (CEV) 
you are to deliver this man to Satan for the destruction of the flesh, so that his spirit may be saved in the day of the Lord.[a] 1 Corinthians 5:5 (ESV)

Hopefully my survey depicts for you the “Saver” as he is meant to be seen and makes explicit his constant objective. He still wants to save you, as only he can. (Almost all of the over 100 uses of the word relate to Jesus and his saving activity - which includes "healing" he did.)

Below I include a (repeated) table listing the occurrences of the verb “to save” (as it does the connected nouns).

Other Greek words (or word families) current in NT about healing or healer? They are found:
• θεραπεύω (therapeuō) "serve, heal" etc: Matt 4:23, 24; 8:7,16; 9:35; 10:1,8; 12:10,15,22; 14:14; 15:30; 17:16,18; 19:2; 21:14; Mark 1:34; 3:2,10; 6:5,13; Luke 4:23,40; 5:15; 6:7, 18; 7:21; 8:2, 43; 9:1,6; 10:9; 13:14; 14:3; John 5:10; Acts 4:14; 5:16; 8:7; 17:25; 28:9; Rev 13:3,12. Also (noun) Matt 24:45; Luke 9:11; 12:42; Rev 22:2 (used for “household” in Matt 24 & Luke 12)
• ἰάομαι (iaomai) "cure", "make whole", etc: Matt 8:8,13; 13:15; 15:28; Mark 5:29; Luke 5:17; 6:18,19; 7:7; 8:47; 9:2,11,42; 14:4; 17:15; 22:51; John 4:47; 5:13; 12:40; Acts 9:34; 10:38; 28:8,27; Heb 12:13; James 5:16; 1 Peter 2:24
• ἰατρός (iatros) Matt 9:12; Mark 2:17; 5:26; Luke 4:23; 5:31; 8:43; Col 4:14
• ὑγιαίνω (hygiainō) "be sound/well", etc: Luke 5:31; 1 Tim 1:10; 6:3; 2 Tim 1:13; 4:3; Titus 1:9,13; 2:1,2; 3 John 2. ὑγιής (hygiēs) Matt 12:13; 15:31; Mk 5:34; John 5: 6, 9, 11,14,15; 7:23; Acts 4:10; Titus 2:8

Some striking words from Isaiah chapter 45 (OT) - long before the time of Jesus:

18 For thus says the Lord,
who created the heavens
    (he is God!),
who formed the earth and made it
    (he established it;
he did not create it a chaos,
    he formed it to be inhabited!):
I am the Lord, and there is no other.

22 Turn to me and be saved,
    all the ends of the earth!
    For I am God, and there is no other.

Note what is said of the LORD in Isaiah 45! That same "save" word is used in the Greek OT (LXX) for verse 22. Today, as you can see above, Jesus invites us to look to him.
There is of course a substantial OT background to "save"; for example NRSV lists 194 occurrences of the English word. (I have not researched the words used in the Hebrew and Greek Bibles.)

Resources: In researching the word occurrences I gained considerable on-line help from and (the latter especially for Strong's numbers, etc).

Scripture quotations marked (CEV) are from the Contemporary English Version Copyright © 1991, 1992, 1995 by American Bible Society, Used by Permission.
Scripture quotations marked (ESV) are from the ESV® Bible (The Holy Bible, English Standard Version®), copyright © 2001 by Crossway, a publishing ministry of Good News Publishers. Used by permission. All rights reserved.
Scripture quotations marked (NIV) are taken from the Holy Bible, NEW INTERNATIONAL VERSION®, NIV® Copyright © 1973, 1978, 1984, 2011 by Biblica, Inc.® Used by permission. All rights reserved worldwide.
Scripture quotations marked (NLT) are taken from the Holy Bible, New Living Translation, copyright ©1996, 2004, 2007, 2013 by Tyndale House Foundation. Used by permission of Tyndale House Publishers, Inc., Carol Stream, Illinois 60188. All rights reserved.
Scripture quotations marked (NRSV) are from the New Revised Standard Version Bible, copyright © 1989 National Council of the Churches of Christ in the United States of America. Used by permission. All rights reserved.

Note: Apologies for formatting distractions in this post. I hope I have (progressively) removed them.