Cavon Baker secured the victory on a defensive rebound and pair of free throws with nine seconds left. Baker finished with 20 points and the 10th-seeded Cougars, winners of four straight, will take on No. 2 seed Tulsa in the quarterfinals on Friday.
Houston (13-18) led 57-45 on Danrad Knowles's jumper with 5:24 left, but seventh-seeded Tulane (15-16) answered with seven straight points and pulled within 64-60 with 22 seconds left. But Kajon Mack missed the second of two free throws and Jonathan Stark and Louis Dabney both missed follow-up chances.
After Tulane tied it 36-all midway through the second half, the Cougars pulled away with a 15-3 run.