Joseph Young scored 14 points and Jherrod Stiggers scored 11 for the Cougars (13-6, 2-4 Conference USA). The game had seven lead changes, but Houston ran off seven straight points to lead 61-55 with 2:26 left.
UAB (8-12, 0-5) closed the margin to 62-61 with 11 seconds left after Jordan Swing made a 3-pointer. Young and Danuel House each hit a pair of free throws to seal the win for the Cougars.
Swing led the Blazers with 20 points and seven assists. Terence Jones scored 14 for UAB.