Thompkins went 11-of-16 from the field and he paced the Bulldogs with seven rebounds. Travis Leslie added 14 points, seven in each half, for Georgia.
Eastern Kentucky, which fell to 5-8, got 15 points from Jeff Johnson off the bench and 12 points and nine rebounds from Spencer Perrin.
Saturday's contest was close throughout, including a 47-47 tie with 7:17 remaining. Thompkins scored the next six points to push the Bulldogs to a 53-47 lead with 5:03 to go. Georgia outscored the Colonels 11-10 to close the game.
The Bulldogs will open Southeastern Conference play as they entertain Kentucky on Jan. 8 at 4 p.m.