How were the Old Testament saints saved?

The OT saints were saved the same way the New Testament saints were/are saved, by faith.

For what does the Scripture say? "And Abraham believed God, and it was reckoned to him as righteousness." ⁴Now to the one who works, his wage is not reckoned as a favor, but as what is due. ⁵But to the one who does not work, but believes in Him who justifies the ungodly, his faith is reckoned as righteousness, ⁶just as David also speaks of the blessing upon the man to whom God reckons righteousness apart from works: ⁷"Blessed are those whose lawless deeds have been forgiven, and whose sins have been covered. ⁸"Blessed is the man whose sin the Lord will not take into account" (Rom. 4:3-8).

As you can see, the Bible tells us that Abraham was justified by faith (see Rom. 5:1 and Eph. 2:8-9). That is, his faith is reckoned as righteousness, v. 4 above. They were saved by faith in the Messiah in whom they were trusting. Only, for them it was a trust in the future Messiah. They knew He was coming as had been prophesied.

Also, the Holy Spirit was there in the OT times the same as the NT times. Consider Psalm 51:11, "Do not cast me away from Thy presence, And do not take Thy Holy Spirit from me."

God did not change how He saved people in the New Testament. It has always been by faith. In the case of the OT people, they looked ahead in time to the Messiah. We look back to Him and see the cross.