In overtime, Kabongo's long 3-pointer gave Texas its first lead since midway through the first half. He followed it with a free throw and a tough layup that put Texas up 83-81.
Sheldon McClellan made two more free throws and Ioannis Papapetrou made a long 3-pointer from the left wing to put Texas ahead 88-84 with 48 seconds left.
Osby scored 31 for Oklahoma (18-9 overall, 9-6 Big 12).
McClellan scored 18 for Texas (13-15, 5-10), including three free throws with 9.5 seconds left in regulation. Osby missed one of two free throws on the other end, setting up Kabongo's desperation shot that tied it at 77.
“I kind of pulled my hand back,” Osby said. “I didn’t follow through. ... I kind of rushed it ... I guess that’s what went wrong.”
Steven Pledger scored 18 for Oklahoma on six 3-pointers but made just one after halftime. The Sooners have lost eight in a row in Austin.
College basketball analysts said before the game that a loss to Texas would not knock the Sooners from the NCAA Tournament, but if they do not defend their home court for the final two home games and beat TCU on the road, the Sooners could be, once again, left out of March Madness.
The Associated Press contributed to this report