TaShawn Thomas added 22 points and six rebounds, and L.J. Rose chipped in 11 points and six assists to lead the Cougars (1-0), which shot 52 percent from the field and hit on 6 of 14 from behind the arc.
Thomas finished 10 of 15 shooting for Houston, while House hit 3 of 5 from behind the arc.
Emani Grant scored 19 points on 9 of 11 shooting, and Joel Wright chipped in 18 points and eight rebounds for the Bobcats (0-1). Phil Hawkins added 10 points for Texas State, which shot 44.8 percent on the game but struggled in the second half, hitting just under 38 percent from the floor.
The Bobcats had cut the lead to 73-70 in the final minute, but House put the game away for Houston, nailing a 3-pointer with 22 seconds to push the lead to six.
Houston opened up its largest lead at 64-52 with 11:27 remaining in the second half on a jumper by Thomas, but the Bobcats would not go away.
Texas State used a 10-1 run over a five-minute span in the second half to cut the lead to three at 68-65 on a Gant jumper with 4:37 left.
After Houston opened the lead back up to five, PhilHawkins hit a 3-pointer with two minutes remaining to close the gap to 72-70.
Rose hit a free throw to open the lead to 73-70 with 1:56 left, but Texas couldn't cut into the lead as Joel Wright turned it over with 1:23 left and House got a rebound on Houston's missed shot before his three.
The Cougars used an 18-3 run over a four-minute span of the first half to open up a 40-29 lead with 4:51 left in the half. House scored nine points in the run and had 17 at the half as Houston held a 46-41 lead at the half.