Understanding how people in the Old Testament were saved can be confusing, given the sacrificial system and the fact that Jesus had not yet died on the cross. However, God did not employ a different means of salvation in the Old Testament compared to the New. Salvation has always been a gift of God’s grace obtained through faith in Him.
To put it simply, Old Testament believers were saved by trusting God to send the promised Messiah (Jesus) to atone for their sins, which was a futuristic event prophesied in the Scriptures (see Isaiah 7:14, 9:6, and 53:3-7; Zechariah 9:9, and Psalm 22:16-18). God simply instituted the sacrificial system as a foreshadowing of what the Messiah would ultimately accomplish once and for all on the cross. Therefore, Old Testament believers looked ahead to the cross, trusting God by faith for their salvation.
New Testament believers are saved by trusting God for having already sent the Messiah (Jesus) to atone for their sins on the cross, which is recorded in the Scriptures (see John 3:16-17 and 14:6, Acts 4:12, 1 John 5:10-12). Therefore, New Testament believers look back to the cross, and all Jesus accomplished, trusting God by faith for their salvation.
I hope this information helps you understand how Old Testament believers were saved before Jesus’ death on Calvary.