Moore was 10 of 15 from the field for the Blazers (6-1), who are 5-0 all-time against the Gamecocks (2-6).
The game was tied seven times in the first half, with Jacksonville State taking a 31-28 halftime edge.
UAB pulled ahead by five with 7:50 to go, but the Gamecocks knotted it again at 48 with 3:17 left. Johnson then had a layup, a three-point play and four free throws to seal the win.
Johnson, who leads the country in assists per game (9.7), had five assists.
The Gamecocks won the battle of the boards 36-31, but the Blazers had a 34-16 advantage in the paint.