Old Testament saints were saved through the PROMISE of the Messiah.
Jesus says in John 8:56- Your father Abraham rejoiced to see my day: and he saw it, and was glad.
Romans 4:3- For what saith the scripture? Abraham believed God, and it was counted unto him for righteousness.
Romans 4:6,7- Even as David also describeth the blessedness of the man, unto whom God imputeth righteousness without works, Saying, Blessed are they whose iniquities are forgiven, and whose sins are covered.
How are all of mans sins covered? Through the blood of the lamb, Jesus Christ, if they accept God's sacrifice of the Christ and decide to live a godly life.
Jesus Christ fulfilled the laws of the OT, which Isreal followed. The saints after Jesus no longer live by the law, but by grace. We are to follow the two great commandments (Matthew 22:37, Mark 12:30, Luke 10:27) which are:
1) Love God with all your heart, mind and soul
2) Love your neighbor as you love yourself