How were men saved prior to Sinai, under the Mosaic law and after the cross?

This is sometimes called the "antiquity of the gospel" -- the idea that only one gospel has ever saved anyone prior to Sinai, under the Mosaic law and after the cross. Men have always been saved through grace by faith in Jesus Christ. In every age, there has only been one gospel that could save anyone.

Prior to Sinai:

And the scripture, foreseeing that God would justify the heathen through faith, preached before the gospel unto Abraham, saying, In thee shall all nations be blessed. So then they which be of faith are blessed with faithful Abraham.
Galatians 3:8-9

The Angel [Jesus] which redeemed me [Israel] from all evil ....
Genesis 48:16

Under the Mosaic law:

Search the scriptures [the Old Testament]; for in them ye think ye have eternal life: and they are they which testify of me [Jesus] .... For had ye believed Moses, ye would have believed me: for he wrote of me. But if ye believe not his writings, how shall ye believe my words?
John 5:39 & 46

Men and brethren, let me freely speak unto you of the patriarch David, that he is both dead and buried, and his sepulchre is with us unto this day. Therefore being a prophet, and knowing that God had sworn with an oath to him, that of the fruit of his loins, according to the flesh, he would raise up Christ to sit on his throne; He seeing this before spake of the resurrection of Christ, that his soul was not left in hell, neither his flesh did see corruption.
Acts 2:29-31

After the cross

Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.
Acts 4:12

Because this is intended to be a brief explanation, we have included just a few of the biblical references that illustrate this fact. It is important to understand that the same gospel of Christ has always been the only means by which people could be saved and have eternal life.

Nevertheless, many evangelical ministers are confused about this issue. Some maintain that a different plan of salvation applied to those in the Old Testament or that they never knew salvation. That is simply not true. The gospel of Jesus Christ is the only way that people could ever be saved.