Bruce Beckford added 11 points and seven rebounds to help North Carolina A&T (15-15, 8-7) win its fifth straight home game.
Trailing 33-18 at halftime, Savannah State fought back to tie the game at 52 on Deven Williams' 3-pointer with 2:41 to play.
After the Aggies went up 56-52 on a pair of free throws from Powell and Middleton each, the Tigers cut the deficit one, 56-55, behind Rashad Hassan's free throw and Preston Blackman's layup with 28 seconds left.
Middleton made three free throws in the final 30 seconds and Hassan missed a layup for the Tigers with 6 seconds left to seal the victory for the Aggies.
Blackman scored 17 points for the Savannah State (17-13, 10-5).