How were people in the Old Testament saved? According to Romans 3:20, they weren’t saved through the law—so it must have been by faith.

Good questions. Before the Fall, God’s standard for salvation would have been perfect obedience to His law. Of course, until this time, the actual need to be “saved” would have been unnecessary. After Adam sinned, God’s standard for salvation did not change. Yet, He provided a substitute of one who perfectly obeyed this law, his son, Jesus, to fulfill this standard. And, as you said in your question, people are saved through faith in Jesus Christ alone. 2 Corinthians 5:21 says, “God made him who had no sin to be sin for us, so that in him we might become the righteousness of God.” As made evident in Romans 7, no man other than Jesus is capable of keeping the law. The law is meant to drive us to Jesus for our righteousness.
Using Abraham as an example, “Abram believed God, and he credited it to him as righteousness” Genesis 15:6, scripture states that Abraham was saved by faith alone. God established his covenant with Abraham, and in Genesis 17, God gave him an outward sign of this covenantal relationship through the act of circumcision. Yet, salvation was by grace alone. Although Abram responded to God’s call, it was God who made the call and established his covenant with man. Salvation is from the Lord alone.
Now, why did people in the OT make sacrifices and keep the law so extensively? The Mosaic law set forth blessings for obedience to the law and cursings for disobedience (much like the fatherly discipline outlined in the NT). Yet, obedience to the law was not a means of earning one’s salvation. Plus, God established 5 different sacrifices for his people in the OT. These included one for guilt, one for sin, and a big annual day of sacrifice called the Day of Atonement. During each sacrifice, the guilt of sinners was symbolically transferred to an animal, which would be punished by either death or exile from the land. These animal sacrifices foreshadowed or anticipated the future sacrificial act of Jesus. These sacrifices, though, rather than save the people, brought the people back into a right relationship with God, much like our confessing our sin to God does today. In other words, these sacrifices did not save the people, but brought them back into fellowship with God, and pointed to the ultimate sacrifice, once for all, that God would provide for His people through His only Son, Jesus Christ.
Finally, the Pharisees were the best example of a group of people who kept the law to the letter. I believe they actually began this practice to please God as they were especially concerned with ceremonial purity. Although this act of strictly keeping the law began as a response to justification by faith, this appears to have changed into legalism and self-righteousness. Unfortunately, because of this change, many of the Pharisees and their followers missed their need for the redeeming grace of Jesus Christ.

-Thanks,
The Young's