Thompkins, who missed the first four games with a high ankle injury, looked as good as he had all season in the first half, when he scored 14 points and Georgia led by as many as 15.
But he took awhile to get going in the second half, and UAB rallied to take several leads. Then with 14.9 seconds left, when teammate Gerald Robinson hit a driving layup to give the Bulldogs a three-point lead, Thompkins got hurt and stayed down on the court for about a minute.
Stegeman Coliseum went quiet when it realized the preseason SEC player of the year was hurt again. But Thompkins walked off on his own, and said afterwards he landed on his hip badly and would be fine.
"Colby just told me to lay there for a second, just to make everyone worry a little," Thompkins joked, referring to athletic trainer Colby Pohlmann.
It was a solid win for Georgia (5-2), which was running out of chances to earn nonconference quality wins. UAB (6-2) figures to be a contender for the NCAA tournament, just like the Bulldogs.
"We knew this would be a real test," Georgia head coach Mark Fox said. "I think they're (the Blazers) a team that's going to win 20 games. And if they stay healthy they'll have a good year."
Georgia put together as good a half as it has all season in the first 20 minutes, jumping out to leads of 16-4 and 36-21. It dominated the rebounding margin in the first half, with more offensive rebounds (11) than UAB had total.
But things reversed themselves in the second half. When UAB got three offensive rebounds on one possession, it hit a 3-pointer to tie it at 51. UAB took its first lead at 57-56 with just under four minutes left.
Then it had the ball with a chance to expand it – but Leslie cut in to make a steal. On the other end, Trey Thompkins made a layup, on his second basket of the half.
Leslie only had four points, but he had seven rebounds and that critical steal.
"Travis is becoming a great defender," Thompkins said. "We pick on him every day, so eventually he'll get tired of it and start playing some defense."
Dustin Ware had another steal on the possession after Leslie. Thompkins followed each with a basket.
But the Bulldogs were still clinging to a one-point lead in the final minute, and had possession with 47 seconds left. UAB elected not to foul, and Fox elected not to call a timeout.
Robinson, left to his own devices, made a driving layup a few seconds before the shot clock would expire. Now up three, Fox elected not to foul, UAB's Jamarr Sanders missed an open 3, Robinson rebounded, was fouled, and hit both free throws.
It was yet another close victory for Georgia, which also beat Manhattan by three, Saint Louis by two and Mississippi Valley State by two.
"I hope we didn't have to keep doing it like this," Ware said. "But we'll take it any way we can get it."