Were the Old Testament Saints Saved by Faith in the Coming Messiah?

Yes. Isaiah and other books prophesy about how the Messiah would suffer and die and be glorified, so the Old Testament prophets, and those who believed them, looked forward to the coming Christ. The Old Testament Scriptures contained, for them, the gospel message, and faith in that message was how people got saved back then, even though Jesus had not died yet.

The Bible says:
"Of this salvation the prophets have inquired and searched carefully, who prophesied of the grace that would come to you, searching what, or what manner of time, the Spirit of Christ who was in them was indicating when He testified beforehand the sufferings of Christ and the glories that would follow. To them it was revealed that, not to themselves, but to us they were ministering the things which now have been reported to you through those who have preached the gospel to you by the Holy Spirit sent from heaven—things which angels desire to look into." (1 Peter 1:10–12)

So the “prophets...who prophesied” longed for the arrival of an era of grace. The “Spirit of Christ” within them was witnessing through them and to them before the work of Christ had occurred, giving them a gospel message that talked about the coming, sufferings, and glory of the future Messiah. So even before the completion of the New Testament, the Old Testament served as the Scripture for Israel, and contained the gospel message.

Acts 3:18 says, "But those things which God foretold by the mouth of all His prophets, that the Christ would suffer, He has thus fulfilled." So the gospel theme was in the Old Testament. It continues through verse 24 and says, in part, "...Jesus Christ, who was preached to you before... which God has spoken by the mouth of all His holy prophets since the world began."

So the gospel message of salvation was preached in the Old Testament, and faith in that gospel message is how people got saved, even though Jesus had not died yet. Paul explained in Romans 4 that salvation has always been, and will always be, by God’s grace, and received through faith alone. Genesis 3:15 promised that Someone would come to clear up the sin problem created by our first father Adam. As the Seed of the woman, He would be the one to battle and defeat the serpent. Even Abel understood the nature of a bloody sacrifice and the death of a substitute, and because of his faith in God, he was regarded by God as righteous (Hebrews 11:4). God has just one covenant of grace, not two, which was promised right after the fall in Genesis 3:15 proclaiming that the seed of the woman would crush the seed of the serpent.