House came off the bench to sink 5-of-8 shots, including three from 3-point range, and give the Cougars (14-14, 6-9 American) back-to-back wins.
Jherrod Stiggers added 14 points for Houston. The Cougars outscored the Tigers (21-7, 10-5) at the free-throw line 21-9.
Michael Dixon scored 19 points for Memphis and Shaq Goodwin doubled up with 16 points and 10 rebounds, The Tigers had won five of six coming into the game at Houston.
Houston led 36-33 at halftime, but fell behind early in the second half. The Cougars reversed the momentum with House hitting a 3-pointer with 2:20 to play to give Houston a 68-63 edge. The lead held up as the Cougars hit nine free throws down the stretch.