Seth Gearhart had 15 points and six boards and Bishop Mercy added nine points for Rice (6-13, 3-4).
L.J. Rose hit a jumper to push the Houston lead to 14, its largest of the game, with 3:05 left in the game. The Cougars finished the final 1:24 of the game 5 of 6 from the free throw line, four from Barnes, to keep the lead in double digits.
Houston forced 15 Rice turnovers, while only coughing it up five times.