"Isn't that why they call it March Madness? Because anything can happen," UTEP coach Tony Barbee said.
"This team has been resilient, tough-minded all year long, and we really bounced back."
UTEP will play Tulsa, which upset No. 2 seed UAB 78-68 in overtime, in the semifinals Friday.
No. 3 seed Houston (22-9) came in looking to reach the semifinals for a third straight year and a victory that could strengthen a case for an NCAA berth. But the Cougars couldn't hold onto an early lead or late after Robert McKiver tried to rally Houston by scoring 14 of his 24 points in the final 6:18.
McKiver almost single-handedly lifted the Cougars to victory, taking over toward the end. His two free throws with 20.6 seconds remaining put Houston up 77-76. Jackson answered with his jumper, then McKiver lost the ball driving to the basket as he was guarded by UTEP guard Julyan Stone.
"We've done that for two years," Barbee said of guarding and stripping McKiver. "It's frustrated McKiver playing us because so much of his game is scoring the ball. If you take the ball out of a scorer's hands like that, it frustrates them. That's been our strategy ever since we played them. I didn't see any reason to change coming into today."
Houston coach Tom Penders refused to comment on the play.
"I've been doing this for 34 years. I've never said that before, but I have no comment on that play," Penders said.
Asked about the runs in the game, Penders called it a great game and great comeback by his Cougars.
"We just kept coming back. Whoever had the ball at the end or was going to get the call ... that was how the game was going to be decided," Penders said.
Jackson hit the final two free throws with 2.5 seconds left, and McKiver's desperation heave hit off the backboard after the buzzer.
Jackson finished with 23 points for UTEP. Marvin Kilgore added 18 -- all in the first half -- Tavaris Watts 12 and Randy Culpepper 10.
Kelvin Lewis added 21 with 19 from Dion Dowell for Houston.
Houston opened the second half with a 12-4 run and went up 47-46 on a pair of free throws by Dowell. That didn't last long as Culpepper, who hit seven 3s in UTEP's 71-49 first-round win over SMU, hit his first 3 of the game that put the Miners back up 49-47 with 12:10 left.
The Miners took control early despite missing their first six shots when Kilgore got hot. He scored eight of 11 straight points as UTEP took a 38-29 lead, and a Jackson jumper left the Miners up 42-35 at halftime.
"They were playing back off of me," Kilgore said. "They were giving me open shots. I hit one, and then I hit two, and I was focused. So, I just kept shooting, and the ball kept going in."