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From a Reformed perspective, we look at the Old Testament promises, and the respondents' trusting God as what qualifies them as righteous and grants salvation.
God chose Abraham and made a covenant with Him (a series of 3 promises you can read in Genesis 21). Abraham believed God, and it was credited to him as righteousness.
Isaac was the son of the promise. He was obedient and faithful to his father and God by willingly laying on the altar at Moriah (Abraham was a really old dude at this point, and it's pretty unreasonable to assume that Isaac was powerless to at least escape this sacrifice, if not overpower Abraham).
Jacob was the son of the promise. His act of faith was recorded with his actions on his deathbed, that he blessed Joseph's sons with the promise of God.
Joseph was the son of the promise. His faithful acts delivered Israel in a time of famine, and lead to Moses leading the people out of Egypt so that God would display His power to His people.
This list goes on. It isn't that each of these men was sinless and trusted Christ, it's that each one was sinful and trusted God in His word that He would bless them (and no small part of that blessing is that they have been included in the bloodline of the Messiah).
Over time, we begin to understand more of the promise. The author wrote concerning this in 2 Samuel 7:12-17. This is a continued promise that God will establish David's kingdom forever and will claim father-hood over this line of people. Isaiah later writes concerning that God's people would "know His name" and God would grant salvation to all ends of the Earth through the Messiah.
Today, it has been revealed that Christ was the fulfillment of the promise to Abraham and that being a "son of the promise" was not fleshly, but spiritual, trusting God. Because it has been revealed that Christ was the fulfillment of the promise, by placing faith in Christ, you trust that God has taken care of your sins on His own account and you are adopted as a son of God and are covered by the promise that God made to Abraham. You are saved the same way that Abraham was saved: by believing God (and, as James 2 would add, demonstrating that belief in your actions).