Devonte Boykins' 3-pointer with six seconds left reduced North Carolina A&T's deficit to 88-85 before Rogers' dunk at the buzzer ended the game.
Ian Kinard made 1 of 2 free throws and South Carolina State led 87-77 with 22 seconds left.
Raymon Pratt made it a seven-point contest with a 3-point play with 12 seconds remaining before Kinard fouled him to send Pratt back to the foul line and he made both. Rogers made 1 of 2 free throws for an 88-82 lead before Boykins' 3.
South Carolina State (9-17, 5-6 Mid-Eastern Athletic Conference) got 13 points from Janai Rayner-Powell and 11 from James Richardson. South Carolina State led by double figures most of the second half.
Davaris McGowens led North Carolina A&T (15-10, 8-2) with 26 points and Boykins scored 13.
XAV VS AUB
Georgia reaches Cayman Islands Classic semis
Rayshaun Hammonds scored a career-high 31 points, Nicolas Claxton had 22 points, 11 rebounds and four assists, and Georgia beat Illinois State 80-68 on Monday in the second game of the Cayman Islands Classic.
Tulane beats South Dakota State to start Gulf Coast Showcase
Caleb Daniels scored 24 points, Shakwon Barrett added 19 and Tulane beat South Dakota State 84-80 on Monday to start the Gulf Coast Showcase.