According to Galatians and Romans, Abraham was saved by faith just like we are saved by faith. A common question is "Did Abraham have to believe in Jesus the same as we do today?" The answer is both yes and no.
The object of saving faith in the OT was the covenant promise of God anticipating Christ; In the NT, it is the covenant promise of God fulfilled in Christ. In either case, the object of saving faith is in fact Christ.
Abraham was saved by believing the word of the promise-making and keeping God. These were not just any promises. They were specifically the promises of a Kingdom (King, People, and Land) in fellowship with God forever. These promises were always and only intended to be fulfilled in Jesus Christ. When Abraham believed God's promise, he may not have been aware of exactly how God would keep his promise. But inasmuch as the promises and their fulfillment in Christ were ordained before the foundation of the world, to believe the promise was ultimately to believe in Christ. So Abraham believed in Christ by virtue of believing in the promise.
Having begun by the Spirit, are you now being perfected by the flesh? Did you suffer so many things in vain—if indeed it was in vain? Does he who supplies the Spirit to you and works miracles among you do so by works of the law, or by hearing with faith—just as Abraham “believed God, and it was counted to him as righteousness”? Know then that it is those of faith who are the sons of Abraham. And the Scripture, foreseeing that God would justify the Gentiles by faith, preached the gospel beforehand to Abraham, saying, “In you shall all the nations be blessed.” So then, those who are of faith are blessed along with Abraham, the man of faith. For all who rely on works of the law are under a curse; for it is written, “Cursed be everyone who does not abide by all things written in the Book of the Law, and do them.” Now it is evident that no one is justified before God by the law, for “The righteous shall live by faith.” But the law is not of faith, rather “The one who does them shall live by them.” Christ redeemed us from the curse of the law by becoming a curse for us—for it is written, “Cursed is everyone who is hanged on a tree”—so that in Christ Jesus the blessing of Abraham might come to the Gentiles, so that we might receive the promised Spirit through faith. (Galatians 3:3-14 ESV)