Taishaun Johnson had 17 points for the Jaguars (7-17, 5-8).
Georgia State used an 11-0 run to build a 41-34 lead with 13:14 to play but the Jaguars followed with a 7-0 to tie it. A 9-2 run rebuilt the lead to 56-48 with 4:24 to go and South Alabama got no closer than five, on a Ken Williams 3 with 58 seconds left.
Ryan Green, Ware and Hunter each made a pair of free throws to secure the win.
Hunter made all seven of his free throws, grabbed five rebounds, and added five assists, three steals and three blocks.