Jones surpassed the 1,000-point milestone for his career.
Fahro Alihodzic chipped in 12 points for the Blazers (8-6).
The Eagles committed 23 turnovers, and Georgia Southern was 5 of 6 from the foul line to UAB's 16 of 21.
Neither team led by more than four points in the first half, with the Blazers ahead 25-22 at the break.
Jordan Swing made a 3-pointer 2:10 into the second half to give UAB a 30-22 advantage. Bussey's layup closed it to 34-31 with 13:14 left, but Rucker made a layup for a 59-52 advantage at 1:15 to seal the win.
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