Eaves was 10-of-16 shooting and made all four of his free-throw attempts.
Ty Soloman hit a 3-pointer to give SC State a 60-58 lead with 1:27 left and Eaves had a 3-point play for a five-point advantage with 57 seconds left. Devin Martin answered with a 3-pointer but Soloman made it a two-possession game again after a layup.
After a UMES turnover, Ed Stephens hit two free throws to extend SC State's lead to 67-61 with 18.5 seconds left.
Daryll Palmer had 10 points, five rebounds and four blocks for SC State (10-10, 5-1 Mid-Eastern Athletic Conference), which had just 11 wins last season.
Dominique Elliott led Maryland-Eastern Shore (4-16, 1-4) with 19 points, and Martin finished with 15.