The sixth-seeded Bearcats (25-10) jumped out to a 16-2 lead while Texas opened the game 1 of 14 and hit 16 percent for the entire first half.
After trailing 31-17 at halftime, the Longhorns attacked, hitting five of their first seven from 3-point range.
With 3:44 to play, Texas' Jonathan Holmes hit a layup to tie it at 52, but the Longhorns would get no closer. Gates hit a shot over Clint Chapman's head with 1:11 to give Cincinnati a 58-52 lead, and the Bearcats hit five free throws in the final 36 seconds.
J'Covan Brown led No. 11 seed Texas (20-14) with 19 points, and Chapman had 10 points and 14 rebounds.