There is a misunderstanding of the Old and New Covenants, but it is in the sense that one is works based and the other is not.
On the contrary, they are both faith based, as Scripture attests. Read Heb 11 and see annotated therein, the list of the Old Testament faithful, who obeyed and did the works of God, by their faith. They are put there as an example for us.
Hebrews 11:1 Now faith is the substance of things hoped for, the evidence of things not seen.
2 For by it the elders obtained a good report.
3 Through faith we understand that the worlds were framed by the word of God, so that things which are seen were not made of things which do appear.
4 By faith Abel offered unto God a more excellent sacrifice than Cain, ...
11 Through faith also Sara herself received strength to conceive seed, and was delivered of a child when she was past age, because she judged him faithful who had promised.
12 Therefore sprang there even of one, and him as good as dead, so many as the stars of the sky in multitude, and as the sand which is by the sea shore innumerable.
13 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.
14 For they that say such things declare plainly that they seek a country.
15 And truly, if they had been mindful of that country from whence they came out, they might have had opportunity to have returned.
16 But now they desire a better country, that is, an heavenly: wherefore God is not ashamed to be called their God: for he hath prepared for them a city.
Those who have enough faith, will do the works which God wills for them to do. Old Covenant or New.
James 2:18 Yea, a man may say, Thou hast faith, and I have works: shew me thy faith without thy works, and I will shew thee my faith by my works.
So, what is the difference between the Old and New Covenants? It's illustrated in this parable:
Matthew 20:1 For the kingdom of heaven is like unto a man that is an householder, which went out early in the morning to hire labourers into his vineyard.
2 And when he had agreed with the labourers for a penny a day, he sent them into his vineyard.
3 And he went out about the third hour, and saw others standing idle in the marketplace,
4 And said unto them; Go ye also into the vineyard, and whatsoever is right I will give you. And they went their way.
5 Again he went out about the sixth and ninth hour, and did likewise.
6 And about the eleventh hour he went out, and found others standing idle, and saith unto them, Why stand ye here all the day idle?
7 They say unto him, Because no man hath hired us. He saith unto them, Go ye also into the vineyard; and whatsoever is right, that shall ye receive.
8 So when even was come, the lord of the vineyard saith unto his steward, Call the labourers, and give them their hire, beginning from the last unto the first.
9 And when they came that were hired about the eleventh hour, they received every man a penny.
10 But when the first came, they supposed that they should have received more; and they likewise received every man a penny.
11 And when they had received it, they murmured against the goodman of the house,
12 Saying, These last have wrought but one hour, and thou hast made them equal unto us, which have borne the burden and heat of the day.
13 But he answered one of them, and said, Friend, I do thee no wrong: didst not thou agree with me for a penny?
14 Take that thine is, and go thy way: I will give unto this last, even as unto thee.
15 Is it not lawful for me to do what I will with mine own? Is thine eye evil, because I am good?
16 So the last shall be first, and the first last: for many be called, but few chosen.
The first set of laborers, are those who do not accept the Sacraments. They will work their entire life, but will not be judged until they come to the Judgment Seat of Christ. At that time, those who claim salvation by faith alone, will be Judged by Christ. Whether they are truly saved, they will learn at that time.
The second set of laborers, are those who accept the Sacraments. Those who take them seriously, repent of their sins and do their best to obey God before and after receiving the Sacraments. They receive the Sacraments and are credited with righteousness for believing that God will keep His promises and for hoping in God for their salvation. Judgment comes to them in this life, everytime they approach the Altar:
1 Peter 4:17 For the time is come that judgment must begin at the house of God: and if it first begin at us, what shall the end be of them that obey not the gospel of God?
God only saves those who do His righteous works, whether it be through the Sacraments or at the Judgment Seat. Both Covenants are based on faith. And in both Covenants, faith without works, is dead.