How did those in the Old Testament receive salvation?

I work in a remote area where the people believe they have to keep the 10 Commandments and sacrifice animals in order to be made right with God, per Old Testament days. How can I explain to them the Old Testament saints were saved by grace through faith?

Salvation has always been the same: by faith in the promises of God. To illustrate this truth, Paul says the Old Testament saint, Abraham, was saved by his faith in God’s promises, not by his works:

Rom. 4:1 What then shall we say that Abraham, our forefather according to the flesh, has found?
Rom. 4:2 For if Abraham was justified by works, he has something to boast about, but not before God.
Rom. 4:3 For what does the Scripture say? “ABRAHAM BELIEVED GOD, AND IT WAS CREDITED TO HIM AS RIGHTEOUSNESS.”
Rom. 4:4 Now to the one who works, his wage is not credited as a favor, but as what is due.
Rom. 4:5 But to the one who does not work, but believes in Him who justifies the ungodly, his faith is credited as righteousness,

Faith has always been the means of salvation for all humanity, but faith always has an object. We place our faith in something. Yet God did not reveal to humanity that Jesus was our Messiah (our "something") until a certain point in history, so prior to that point, what did the Old Testament saints believe in?

When Jesus appeared, He became the object of our faith, but prior to Jesus, God still gave lessor revelations of that coming provision. So the object of the Old Testament saint's faith was always the same: the promises of God. The Old Testament saints believed in the promises of God concerning His coming provision for sin even though they did not understand that provision completely. The content of God’s promise were revealed in varying degrees over time, but the object of His promise (i.e., a provision for sin) remained unchanged.

Hebrews says it this way:

Heb. 1:1 God, after He spoke long ago to the fathers in the prophets in many portions and in many ways,
Heb. 1:2 in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world.

Jesus is the culmination of God’s revelation concerning His provision for sin, but earlier generations of saints received only portions of that story. Nevertheless, everyone had the same object for their faith: God’s promises.

For example, Noah believed in God’s promise that an ark would save Noah and his family from a coming flood. Abraham believed God’s promise of a child from his barren wife, etc. Hebrews 11 gives us a record of the faith of many Old Testament saints, and in every case the saint believed in a promise of God, and these promises were portions of the revelation of Christ. So the Lord has always declared that people were saved by faith and not by works.

For a more detailed explanation of these matters, we highly recommend you listen to our entire Romans Bible study.