What assures Christians that they alone are saved? Christians claim that "they are saved by the sacrifice of Jesus." But there are also members of many other faiths who claim that they are saved. For example, apart from affirming Jesus to be the Messiah, the Qurʾān explicitly claims by name that those who believe Jesus as God are disbelievers. In addition, Judaism generally considers Christianity to be idolatry or avodah zarah, since Christianity generally affirms Jesus to be God.

What is the concrete, Biblical basis that assures that it is this specific group of Christians like the modern claimants of Christianity that they are indeed saved, compared to other faiths who also have their own scriptural basis for believing they are saved?
First question to ask is what are Christians being saved from or for? Thankfully someone has saved me time from having to write that answer up (Thanks @Caleb):

What does it mean to &quot;be saved&quot;

Again, I would promote the necessity of relationship with God so that a follower knows the nature of God well enough to know what His desire for us. This is summed up in the famous and oft quoted verse John 3:16:


  16 “For God so loved the world, that He gave His [a]only begotten Son,
  that whoever believes in Him shall not perish, but have eternal life.


Caleb answered what we are saved from, my last quote tells us that God does not want to lose any of us, my next part will explain the part that causes us to be in the firing line of Gods wrath.....

Rom 3:23


  23 for all [a]have sinned and fall short of the glory of God,


This means we have missed the mark, we have FAILED the test, the Law we have broken. When the Law is broken justice is required. God is a just God so cannot simply ignore that we have sinned and grant mercy, otherwise He is not just. We, as humans cannot remove the need for justice, just as we cannot remove the stain of sin, any attempt to do so is futile:

Isaiah 64:6:


  6 For all of us have become like one who is unclean, And all our
  righteous deeds are like a filthy garment; And all of us wither like a
  leaf, And our iniquities, like the wind, take us away.


We cannot base our idea of righteousness as the grading system that God will use, for His standards are far higher than ours.

How do you ascertain the standard of God? Would Mother Teresa make it based purely upon the standard she demonstrated? Would Hitler based upon his actions meet the requirements of Gods standards? No, surely not in the case of Hitler, but also Mother Teresa, as amazing as she was, also fell short of the requirements that would make her salvation possible.

Long story short, man cannot do anything in his own works to gain salvation. Justice IS required by and sought by God: Psalm 94

So, hopefully I have established from a Christian perspective why we are in the firing line of Gods wrath. Now to be sure of salvation in Christ as a Christian:

So, as Christians believe that it is through Christ that salvation is made possible, because our sinful ways and the requirement for justice made outside intervention a necessity, because mere man was unable to undertake this correction for themselves. Jesus was able to satisfy the demand for justice so that all that was required was for the mercy and grace of God to have a chance to win back from Satan the hearts of man.

To be sure of their salvation a Christian needs to have a relationship with God because then you are learning of His nature such as God cannot lie:

Titus 1:2


  2 in the hope of eternal life, which God, who cannot lie, promised
  [b]long ages ago,


He loves us: John 3:16

If then, Christians agree that Paul's words are divinely inspired scripture (and remember God cannot lie) then it is plain and simple for Christians to be certain of their salvation:

Romans 10:9-10


  9 [a]that if you confess with your mouth Jesus as Lord, and believe in
  your heart that God raised Him from the dead, you will be saved; 10
  for with the heart a person believes, [b]resulting in righteousness,
  and with the mouth he confesses, [c]resulting in salvation.


Furthermore, a Christian believes they are being saved so they can live, but they can only live if they do not lead their live according to the flesh, but by the Spirit.

Romans 8:13


  13 for if you are living according to the flesh, you [a]must die; but
  if by the Spirit you are putting to death the deeds of the body, you
  will live.


A Christian knows if they are living by the flesh because they act according to their own desire, pursuing their own agenda, but ultimately not seeking to know God more and more.

So, as a Christian, to be sure of your salvation; do not simply live to an acceptable level (i.e., the Law or to mans standards), but rather live by the Spirit.