Darian Thibodeaux scored 21 points and his pair of free throws with two seconds remaining was the difference in Houston's 76-75 win over Rice on Saturday.
Trailing 75-74, Thibodeaux was fouled and nailed both free throws in the waning seconds to give Houston (15-14, 7-9 C-USA) the win.
Rice (17-14, 8-8) overcame a 13-point second-half deficit to lead 74-73 with 16 seconds remaining on a layup by Arsalan Kazemi, capping a 19-7 run by the Owls that spanned 8:24. Both teams traded a free throw before Thibodeaux's game-winning heroics.
Houston jumped out to an 8-0 lead in the game's first four minutes, but trailed by four at the first half's 8:14 mark. The Cougars concluded the half with six unanswered points to lead 39-34 at the break.
Kazemi led Rice with 19 points.
The Owls shot 51.9 percent from the field to the Cougars' 42.6 percent.