Aldridge scored four touchdowns, including a 33-yard run that put the Cougars (3-3, 2-1 Conference USA) ahead 49-48 with 3:37 remaining. He added a 50-yard TD for the final points two minutes later.
Avery set conference and school records with his receiving total and finished with 427 all-purpose yards. He had two touchdowns.
The Owls (1-5, 1-1) led 48-35 going into the fourth quarter, but Houston quarterback Case Keenum, who replaced starter Black Joseph in the second half, scored on a 2-yard keeper early in the quarter to get the rally going.
Clement threw for a 355 yards, a career high, and three touchdowns and led the Owls with 52 rushing yards. Dillard had 133 receiving yards and a first-quarter touchdown.
Trailing 34-28 just before halftime, Houston had a chance to take the lead with 16 seconds left, but Joseph committed his second turnover when he fumbled on the Rice 1.
Joseph was 11-of-18 passing for 318 yards and three touchdowns in the first half. Avery had 208 receiving yards in the first quarter, a school record for a quarter.
South Carolina football players will wear stickers on their helmets this season to honor the nine people who died in a shooting at Emanuel AME Church in Charleston this past June.
Washington has found its starting quarterback, but coach Chris Petersen said the Huskies will keep that information under wraps until Friday's season opener at No. 23 Boise State.
Joey Counts scored on a 1-yard run on fourth-and-goal with 2 seconds left to give Montana a 38-35 victory over four-time defending FCS champion North Dakota State on Saturday.
Transfer Vernon Adams will be the starting quarterback when Oregon begins its season against Eastern Washington on Sept. 5.
Tim Beckman is threatening legal action after Illinois fired him as football coach Friday amid allegations of influencing medical decisions and pressuring players to play hurt.
LSU offensive coordinator Cam Cameron revealed in a post-practice news conference that he has been treated for prostate cancer but plans to coach this season.