Hicks, who has averaged a double-double over the past 14 games, had 14 points and 12 rebounds for Cincinnati (16-8, 5-5 Big East). Jihad Muhammad had 19 points, Devan Downey 15 - including seven in the final four minutes - and James White had 14.
Louisville, which beat Cincinnati 67-50 two weeks ago, has won just twice in eight games but was coming off an overtime victory over Notre Dame. Cincinnati had lost six of eight.
Jenkins tied the game at 41 before Hicks, who sat out the first 4 1/2 minutes of the second half, returned and the Bearcats regained control.
Louisville made one more run, pulling within 67-66 on a 3-pointer by Dean with 3:40 left. But Downey hit a 3-pointer and drove for a layup, and Muhammad hit a pair of free throws.
Dean hit a 3-pointer for the first basket of the game, but the Cardinals missed their next five shots while Muhammad made a pair of 3's to start Cincinnati on an 18-3 run that gave the Bearcats their largest lead at 18-6.
Louisville chipped away, but Hicks single-handedly kept Cincinnati in front, accounting for all eight of Cincinnati's points during a four-minute stretch.
A pair of free throws by Dean with 1.8 seconds in the first half left pulled Louisville within 1.
Hicks, who came down hard on the play that resulted in Dean's free throws, went to the locker room while Dean was at the line.
Louisville, which hit a season-high 13 3-pointers against Notre Dame, was 6-of-10 from behind the arc in the first half.
Hicks returned to the locker room shortly after the second half started. Louisville's 6-foot-11 David Padgett was able to dominate until Hicks returned with 15:46 to play.