Keenum ran for two scores, including a 1-yard sneak to give the Cougars a 42-37 leadas they rallied from a 10-point deficit.
UTEP (5-6, 4-3) got the ball to the Houston 44 with five seconds left, but Trevor Vittatoe's last second pass dropped into the end zone incomplete to end the game.
Keenum was 41-of-51 passing and threw three touchdowns. Keenum found Kierrie Johnson in the back of the end zone on a 16-yard pass that gave Houston a 36-35 lead with 4:54 left.
On the conversion attempt, Josh Ferguson intercepted Keenum and returned it for two points to hand UTEP a 37-36 advantage.
Houston got the ball back one more time and marched 59 yards for its game-winning drive.
Houston finished with 700 total yards of offense, while UTEP had 464 yards of offense.
The Miners took their largest lead of the game, 28-9, with a 4-yard touchdown pass from Vittatoe to Moturi with 4:37 left in the third quarter.
Houston closed within 12 on a 15-yard touchdown pass from Keenum to Bryce Beall with 3:45 left in the third. The Cougars climbed closer to start the fourth quarter with a 62-yard touchdown pass to Tyron Carrier from Keenum to cut the lead to 28-23.
UTEP extended the lead with a 42-yard touchdown pass from Vittatoe to Adams with 10:37 left in the game to make it 35-23, but Beall cut the lead to five on a 71-yard touchdown run 1:03 later getting two blocks from wide receiver L.J. Castille on the run.
Beall, a freshman running back, finished with 167 yards rushing on 19 carries with one rushing touchdown and he caught eight catches for 96 yards and a touchdown. Beall went over 1,000 yards rushing in the game.
Carrier caught nine catches for 141 yards and a touchdown. Vittatoe was 27-of-45 for 362 yards, and Kris Adams 111 yards and three touchdowns receiving.