Freshman Patrick Patterson added 16 points as Kentucky (10-9 overall, 5-2 Southeastern Conference) moved above .500 for the first time since December.
Kentucky played strong late in the game without Crawford, who suffered cramping in his foot.
Kentucky lost Bradley, a guard, with 8:44 remaining in the first half when the senior, attempting a fast-break layup, was knocked to the court from behind by Georgia's Dave Bliss.
Bradley, who joined warmups for the start of the second half before he was sent back to the locker room, suffered a Grade 2 concussion and a cut on his chin, according to coach Billy Gillispie.
Sundiata Gaines, who led the Bulldogs with 15 points, hit a 3-pointer with 5:19 left to give Georgia its last lead at 54-53, but the Bulldogs managed only one more field goal the rest of the game, a Troy Brewer 3-pointer with 25 seconds remaining.
Georgia (11-8, 2-4) suffered its third loss in a row and had its 11-game home winning streak end.
The Bulldogs, last in the SEC in scoring and shooting in conference games, have been held to fewer than 60 points in four of six SEC games.
"I thought we played really hard today, especially on defense, but we just couldn't capitalize on the opportunities that we had throughout the game," Georgia coach Dennis Felton said.
"We had 17 offensive rebounds, which is a tremendous number, and gave ourselves plenty of chances to build a lead, but we were never able to break through and give ourselves any type of cushion."
Kentucky led only 30-29 at halftime but stretched the advantage to 40-29 with a 10-0 run.
After Gaines gave Georgia its last lead, Crawford answered with his fifth 3-pointer.
Georgia trailed 58-55 when Terrance Woodbury committed a turnover which led to an alley-oop jam by Patterson, giving Kentucky a 60-55 lead with 56 seconds left.
Bliss had 10 points and eight rebounds, helping Georgia to a 40-31 advantage in rebounds.