Thank you for this illuminating question. It is so good that I have two answers.
First, anything good comes from Christ. If the planets rotate on their orbits, that goodness comes from Christ too. From Him all good things come. When an Atheist, Jew, Muslim or pagan does something good, it is Christ working in him.
Second, was the faith effectively imparted to the good worker? Or is the good worker acting on the law written in his heart, the Natural Law but he does not know the faith?
The Gospel of Matthew explains that a good worker who clothed the naked and feed the hungry may not recognize Christ in the person he helps (Mt 25:37f). He may even be surprised to learn that it is Christ Whom he loved.
So we conclude that while no good work is possible without the grace of Christ, it is possible to not recognize the fact and still do good work. That is the second part of the answer.
If not performed through faith in Christ, then they have their rewards already and it was done by themselves independent of Him.
Anybody performing outside of faith through Him, is acting in sin.
In order for Christ to work in the person, they must first take on the mind of Christ and allow Him to indwell within them. Christ is unable to indwell the person, until that person is indwelt by God the Holy Spirit, and this does not occur until something spiritually righteous is found in that person. Faith alone in Christ alone does provide something God finds to be righteous per Rom 3:22. Once this happens, God the Holy Spirit regenerates the human spirit, providing a temple for the indwelling of God in the believer. Once indwelt, and remaining in fellowship with Him, God the Holy Spirit is able to further sanctify the human spirit, which then makes sound Bible doctrine understood by the mind in the soul of the believer. A the believer matures with more doctrine circulating in his soul, he then may begin to take on the mind of Christ.
Man ate from the tree of the knowledge of good and evil in the Garden. He is able to discern and perform good works, but not necessarily by God's Plan or by His Divine Standards unless it is performed in a right way. A good work is righteous by Divine standards, only when performed through faith in Christ.
There are many good works performed everyday independent of faith in Christ, but those will all be burnt up as hay and stubble or dross in the refining fire or defined as good for nothingness.