Rudd also had a go-ahead three-point play with 13.5 seconds left - that made it 85-84 - and two late free throws in the third overtime.
"He was very special tonight," South Florida coach Stan Heath said. "This was probably his best clutch performance."
South Florida made a late, regulation run, getting within 60-58 on Anthony Collins' layup with 14.6 seconds remaining. After A'uston Calhoun converted 1 of 2 free throws, Rudd tied it at 61 on his 3-pointer.
"For me, it's an extra good Christmas," Rudd said.
Crawford scored all of his 27 points after halftime for Bowling Green (5-6). Calhoun added 22 points.
"This was, I guess maybe from a basketball fan, a neutral point of view, a great game," Bowling Green coach Louis Orr said. "I will give our guys credit. They didn't quit playing, but it's tough. A tough, tough game."
The game was delayed 22 minutes with 21.6 seconds to go in the first overtime when an electrical problem knocked out most of the arena's lights.
Crawford had two baskets, including a driving layup, in the final 15 seconds of the first OT that got Bowling Green even at 71.
Crawford had a fast-break layup before Rudd made his second late, tying 3-pointer to tie it at 78 with 27.4 seconds left in the second overtime.
Crawford, who entered averaging 15.7 points, got his first points on a trey 26 minutes into the game that cut Bowling Green's deficit to 43-34. His jumper from behind the arc gave the Falcons a 51-49 lead with 8:25 remaining.
Crawford missed his only shot of the first half 6 1/2 minutes into the session. His fourth 3-pointer made it 60-53.
Brock had 13 points to help South Florida grab a 34-27 halftime advantage. He also had a key shot block just before Rudd's game-deciding three-point play.
Collins, who will have a sore hip evaluated this weekend, had eight points and eight assists.Click here for to share your thoughts with other USF insiders -- get the inside scoop!