The Bearcats (25-10) led by 19 points 4 minutes into the second half but Texas scrapped back and tied the score at 52-52 with 3:46 to go.
Gates converted a feed from Cashmere Wright and Wright, after a Texas 3-point miss, made a layup of his own. The Longhorns missed another long shot and Gates knocked down a jump shot for a 58-52 lead with 1:14 left.
Cincinnati, the No. 6-seed in the East Region, heads into a Sunday third-round contest against third-seeded Florida State, which just got by St. Bonaventure 66-63 Friday.
Texas (20-14) was nearly able to overcome a horrendous start. The Longhorns managed just two points through the first 12 1/2 minutes the game. They missed 11 consecutive shots at one point.
Cincinnati used the Texas lull to build a 16-2 advantage and was up 38-19 with 16:02 left before the Longhorn comeback.
Gates added 10 rebounds to his 15-point effort. Jaquon Parker had 13 points while Dion Dixon scored 12 and Wright 11 for the Bearcats.
J'Covan Brown, who fueled the comeback with 3-point shots, paced Texas with 19 points. Clint Champion had 10 points and 14 rebounds.