South Florida (18-11, 11-5) is seeking its first NCAA Tournament berth since 1992. Cincinnati (20-9, 10-6) saw its three-game winning streak come to an end.
For the sixth consecutive outing, South Florida held an opponent to 20 points under its scoring average. Cincinnati, which shot just 34.6 percent from the field (18 of 52), came in averaging 69.8 points per game.
Collins finished with 14 points for the Bulls, who went up 44-43 on two free throws by Ron Anderson Jr. with 26.3 seconds remaining.
Cincinnati retook the lead, 45-44, on Cashmere Wright's basket with 9.5 seconds left.
Sean Kilpatrick paced Cincinnati with 13 points.
After Collins' go-ahead free throws, Cincinnati had one final opportunity. But South Florida, with two fouls to give before a bonus situation, committed a foul in the frontcourt. With 0.7 seconds remaining, an inbounds pass attempt by Cincinnati's Dion Dixon was deflected by South Florida's Victor Rudd Jr.
South Florida maintained a lead throughout the second half until 5:56 remaining, when Cincinnati's Yancy Gates scored his first points of the game on an alley-oop dunk, giving the Bearcats a 39-37 advantage.
Cincinnati stretched its lead to four points, but South Florida surged ahead on a baseline jumper by Jawanza Poland, followed by a 3-pointer from Toarlyn Fizpatrick.
Wright's jumper with 1:03 remaining put Cincinnati up 43-42, setting up South Florida's final free-throw attempts by Anderson and Collins.