The Bulldogs (12-13, 4-8 Southeastern Conference) won their third straight at home to improve to 11-3 this season at Stegeman Coliseum.
Georgia snapped an 11-game losing streak to SEC Western Division opponents.
Alabama (14-12, 4-8) has lost five of six, but the Crimson Tide took their biggest lead when Mikhail Torrance's two free throws made it 52-39 with 11:39 remaining.
Since beating Georgia last year in Tuscaloosa, the Tide had lost five straight and six of eight to the SEC East.
Thompkins, who finished with a season-high 17 rebounds, banked in a 14-footer with 4:12 left in the game to make it 61-60 and give the Bulldogs their first lead in nearly 13 minutes.