Houston (4-3, 3-0 Conference USA) trailed 35-23 heading into the fourth quarter before Keenum, who surpassed 300 yards passing for the eighth straight game, engineered the comeback.
Trailing 38-36, the Cougars got the ball at their own 15 with 1:58 left.
Keenum, who completed 28 of 36 passes for 404 yards, was 5-of-6 on an eight-play drive. He capped it with his TD pass to Carrier, who made a nice over-the-shoulder catch in the end zone.
SMU (1-7, 0-5) has lost 14 straight conference games and 17 in a row to FBS teams. The Mustangs' only win this season was against FCS school Texas State.
Keenum's 9-yard touchdown run brought the Cougars within 35-30 early in the fourth.
After an SMU field goal, Beall scored on a 1-yard run but Houston's two-point conversion attempt to tie the game failed. The Mustangs were forced to punt on their next possession, setting up the Cougars' game-winning drive.
Mitchell's desperation pass from near midfield on the final play of the game was intercepted.
The Mustangs last won a conference game on Nov. 18, 2006.
SMU jumped out to a 21-7 lead thanks to Mitchell's first TD run of the season and two scoring passes to Emmanuel Sanders, who set the SMU school record with 26 touchdown receptions.
The Cougars crept within 21-17 early in the third. Mitchell tossed a pair of touchdown passes to Robinson, including a 70-yarder on the final play of the third quarter to extend SMU's lead to 35-23.
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After finishing 97th in the FBS in scoring and 75th in passing yardage, Minnesota has fired offensive coordinator Matt Limegrover and passing game coordinator Jim Zebrowski, the school announced.
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Six days after his death from brain cancer, hundreds of people, including Michigan football players, attended a memorial service for the 5-year-old grandson of former coach Lloyd Carr on Sunday.