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.
Bo Levi Mitchell was 29-of-39 for 365 yards and four touchdowns, and Aldrick Robinson had 186 yards receiving and two touchdowns for the Mustangs.
Bryce Beall rushed for 105 yards and three touchdowns and Mark Hafner had 160 yards receiving for the Cougars, who are 3-0 in league play for the first time since 1990.
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.