Describe how people were saved in the Old Testament after the law was given?

In trying to understand Old Testament salvation, we have both the advantage and the disadvantage of having the New Testament. The New Testament is an advantage because we have some interpretive statements as to the purpose and limitations of the law and ways in which God dealt with Israel at this time. However, the New Testament works to our disadvantage because we will find it all but impossible to approach the Old Testament teachings without imposing our New Testament understanding on it. Therefore, we tend to read New Testament truths back into the Old Testament. The goal is to find a proper balance between the two.

I do not have a detailed plan of salvation to give that strictly applies to Old Testament saints. I do not even feel that it is critical for us today to understand in full detail the way they came to God. However, I will gladly give several of the biblical guidelines that will help keep us in safe territory.

First, the law in and of itself did not then and cannot today be a source of eternal life. The law provided two ways for man to approach God through his own works: by obedience to the commandments (Exodus 24:1-8 [1] And he said unto Moses, Come up unto the LORD, thou, and Aaron, Nadab, and Abihu, and seventy of the elders of Israel; and worship ye afar off. [2] And Moses alone shall come near the LORD: but they shall not come nigh; neither shall the people go up with him. [3] And Moses came and told the people all the words of the LORD, and all the judgments: and all the people answered with one voice, and said, All the words which the LORD hath said will we do. [4] And Moses wrote all the words of the LORD, and rose up early in the morning, and builded an altar under the hill, and twelve pillars, according to the twelve tribes of Israel. [5] And he sent young men of the children of Israel, which offered burnt offerings, and sacrificed peace offerings of oxen unto the LORD. [6] And Moses took half of the blood, and put it in basons; and half of the blood he sprinkled on the altar. [7] And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient. [8] And Moses took the blood, and sprinkled it on the people, and said, Behold the blood of the covenant, which the LORD hath made with you concerning all these words.
See All...; Leviticus 18:5Ye shall therefore keep my statutes, and my judgments: which if a man do, he shall live in them: I am the LORD.
See All...; Deuteronomy 27:26Cursed be he that confirmeth not all the words of this law to do them. And all the people shall say, Amen.
See All...) and by the making of sacrifices (Leviticus 1:1-4 [1] And the LORD called unto Moses, and spake unto him out of the tabernacle of the congregation, saying, [2] Speak unto the children of Israel, and say unto them, If any man of you bring an offering unto the LORD, ye shall bring your offering of the cattle, even of the herd, and of the flock. [3] If his offering be a burnt sacrifice of the herd, let him offer a male without blemish: he shall offer it of his own voluntary will at the door of the tabernacle of the congregation before the LORD. [4] And he shall put his hand upon the head of the burnt offering; and it shall be accepted for him to make atonement for him.
See All...; 4:27-35
See All...). But neither of these gave eternal life. Obedience to the commandments could not save man because no man could keep all the commandments (Ecclesiastes 7:20For there is not a just man upon earth, that doeth good, and sinneth not.
See All...; Romans 3:19-20 [19] Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God. [20] Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin.
See All...; 12-14; 8:3
See All...; Galatians 2:16Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ, even we have believed in Jesus Christ, that we might be justified by the faith of Christ, and not by the works of the law: for by the works of the law shall no flesh be justified.
See All...; 3:10-12 [10] The mountains saw thee, and they trembled: the overflowing of the water passed by: the deep uttered his voice, and lifted up his hands on high. [11] The sun and moon stood still in their habitation: at the light of thine arrows they went, and at the shining of thy glittering spear. [12] Thou didst march through the land in indignation, thou didst thresh the heathen in anger.
See All..., 21-22; Hebrews 7:18-19 [18] For there is verily a disannulling of the commandment going before for the weakness and unprofitableness thereof. [19] For the law made nothing perfect, but the bringing in of a better hope did; by the which we draw nigh unto God.
See All...). The making of sacrifices could not save man because the blood of animals could not put away sin. We can see this both in Old Testament teaching (Psalm 51:16For thou desirest not sacrifice; else would I give it: thou delightest not in burnt offering.
See All...; Isaiah 1:11To what purpose is the multitude of your sacrifices unto me? saith the LORD: I am full of the burnt offerings of rams, and the fat of fed beasts; and I delight not in the blood of bullocks, or of lambs, or of he goats.
See All...; Hosea 6:6For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.
See All...) and in New Testament teaching (Hebrews 9:8-10 [8] The Holy Ghost this signifying, that the way into the holiest of all was not yet made manifest, while as the first tabernacle was yet standing: [9] Which was a figure for the time then present, in which were offered both gifts and sacrifices, that could not make him that did the service perfect, as pertaining to the conscience; [10] Which stood only in meats and drinks, and divers washings, and carnal ordinances, imposed on them until the time of reformation.
See All..., 10:1-4
See All..., 11). No man has ever been saved by keeping the law.

Second, even under the law, men came to God ultimately by faith. Even though their faith might be expressed in works (as ours should be), it was still faith that got God's attention. Hebrews 11 is often called the faith chapter in the Bible. The passage gives a list of men and women of the Old Testament who were known for their faith. Several of these lived under the law. In Hebrews 11:13These all died in faith, not having received the promises, but having seen them afar off, and were persuaded of them, and embraced them, and confessed that they were strangers and pilgrims on the earth.
See All... we read this general statement: "These all died in faith, not having received the promises, but having seen them afar off, and were persuaded of them, and embraced them, and confessed that they were strangers and pilgrims on the earth." Evidently, the key that united all of these men and women was their faith and the fact that they died in faith.

But what did their faith do for them? Well, it seems to have imputed unto them (at least in the legal sense) the righteousness of God. I realize that the proof text often used in the New Testament refers to the faith of Abraham, a man who lived long before the giving of the law (Genesis 15:1-6 [1] After these things the word of the LORD came unto Abram in a vision, saying, Fear not, Abram: I am thy shield, and thy exceeding great reward. [2] And Abram said, LORD God, what wilt thou give me, seeing I go childless, and the steward of my house is this Eliezer of Damascus? [3] And Abram said, Behold, to me thou hast given no seed: and, lo, one born in my house is mine heir. [4] And, behold, the word of the LORD came unto him, saying, This shall not be thine heir; but he that shall come forth out of thine own bowels shall be thine heir. [5] And he brought him forth abroad, and said, Look now toward heaven, and tell the stars, if thou be able to number them: and he said unto him, So shall thy seed be. [6] And he believed in the LORD; and he counted it to him for righteousness.
See All...; Romans 4:1-5 [1] What shall we say then that Abraham our father, as pertaining to the flesh, hath found? [2] For if Abraham were justified by works, he hath whereof to glory; but not before God. [3] For what saith the scripture? Abraham believed God, and it was counted unto him for righteousness. [4] Now to him that worketh is the reward not reckoned of grace, but of debt. [5] But to him that worketh not, but believeth on him that justifieth the ungodly, his faith is counted for righteousness.
See All...; Galatians 3:6Even as Abraham believed God, and it was accounted to him for righteousness.
See All...; James 2:23And the scripture was fulfilled which saith, Abraham believed God, and it was imputed unto him for righteousness: and he was called the Friend of God.
See All...). But consider the following passage from Romans.

Romans 3:21But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets;
See All... But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets; 22 Even the righteousness of God which is by faith of Jesus Christ unto all and upon all them that believe: for there is no difference:

This passage states that the righteousness of God without the law was witnessed by the law and prophets. This certainly refers to the imputed righteousness of God on those who believe. Certainly, the Old Testament saints did not have to understand the payment that Christ would make for them on the cross. Neither am I claiming that Old Testament salvation is equal in every respect to that of New Testament saints. It is not. Their sins were not totally cleansed as ours are. They did not have the indwelling of the Spirit as a seal to their faith. They probably did not have the assurance of eternal life, at least not in the way we have it today. However, they accepted the word of the Lord, trusted in Him, and had His righteousness imputed to them because of their faith. This held their judgment in reserve until the coming of Christ and the application of His shed blood "for the redemption of the transgressions that were under the first testament" (Hebrews 9:15And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.
See All...).

I realize that I have not answered all the questions you may have. I am sure that I cannot answer many questions that could be asked. This is a complex subject. Unfortunately, believers are now using Old Testament salvation as another cause of division. This is true even among those who agree on the plan of salvation for today. This is sad. But I hope that my comments will at least be a help to you in your study of God's word.