House scored 19 points, including an important 3-pointer down the stretch, and the Cougars closed with a surge to beat No. 21 Memphis 77-68.
''It's always been like that, ever since middle school, because my coaches and my teammates believe in me,'' House said. ''I'm glad that they do believe in me to take and knock down the big shots. It's not just on me. Each one of our guys can take a shot and knock it down. They just chose me and I really appreciate that.''
TaShawn Thomas added 16 points and seven rebounds for the Cougars, who finished on a 14-5 run to win their second consecutive game. Jherrod Stiggers chipped in with 14 points and L.J. Rose had seven points, seven rebounds and 10 assists for Houston (14-14, 6-9 American Athletic Conference).
The Cougars improved to 14-0 when leading with 5 minutes left. They shot 43 percent from the floor, including 8 of 21 on 3-pointers.
''Really proud of our guys,'' Houston coach James Dickey said. ''They competed. Down the stretch we just gutted it out. Our defense was better. Our rebounding was better. We would have liked to take care of the ball a little better. They got some run-outs, but not as many and not as many easy baskets as they did in Memphis.''
Michael Dixon Jr. scored 19 points and Shaq Goodwin had 16 points and 10 rebounds for Memphis (21-7, 10-5). Austin Nichols added 12 points.
Dixon called it a horrible loss and said it stung.
''Ultimately, I don't think we played any defense,'' he said. ''We were horrible on the defensive end as a team. That decided the game. They were hitting wide-open shots and made a lot of shots. They are a good shooting team. Totally different team at home than on the road and that decided the game.''
With the score tied at 63 with 3:53 remaining, Rose started the Cougars on their game-ending run with two free throws. House hit a 3-pointer before Dixon cut the lead to three with a layup with 1:33 remaining, but that was as close as Memphis got the rest of the way.
The Cougars hit eight of 10 free throws down the stretch in earning their second victory over a ranked opponent for the first time since the 2009-10 season. Houston beat then-No. 17 Connecticut on Dec. 31.
The Tigers shot just under 47 percent but only 3 of 13 from 3-point range and committed 14 turnovers that led to 21 Houston points.
Memphis coach Josh Pastner said his team has been fortunate to win some tight games and this time it caught up to the Tigers. They have close wins against Central Florida, Rutgers and Temple.
''Everyone's good, but you have to give Houston credit because they made some big plays,'' Pastner said. ''They are a good team. We didn't get the job done, which is a total credit to Houston.''
Houston opened the second half with four straight points to extend its lead to 40-33 on two free throws by Rose with 18:57 left, but Memphis answered with a 15-4 run to grab a 50-46 lead on Dixon's 3-pointer with 12:47 remaining. Dixon had eight points in the spurt.
Houston answered with a 9-0 surge to take a 55-50 lead on a second straight 3-pointer by Stiggers with 11 minutes remaining, but the Tigers responded again with six consecutive points to grab a one-point lead with 9:46 left on Nick King's jumper.
''They hit big 3s,'' Pastner said. ''Stiggers hit back-to-back big 3s and House hit a big 3 late. They hit some big shots.''
In a back-and-forth first half, there were seven lead changes and five ties.
Memphis held four-point leads several times, the last time coming with 5:57 left in the first half on Goodwin's dunk, but Houston ended the half on an 11-4 spurt and scored the last six points to take a 36-33 lead at the break on House's four-point play.
Thomas came in needing five rebounds to become the seventh player in Houston history with more than 1,300 points and 800 rebounds in his career, joining Elvin Hayes, Dwight Davis, Greg Anderson, Rickie Winslow, Clyde Drexler and Hakeem Olajuwon.
''It's an honor,'' Thomas said. ''Growing up I never thought I would even be mentioned with those guys, so being able to play at the same school, and being able to put my name alongside those is just an honor. Plus it's my birthday, so I can definitely say that it's one of the best birthdays I've ever had.''