Remarkably, the only points of the third quarter came via Houston's safety after Hill was sacked on consecutive plays.
BYU then took a 41-40 lead on Hill's 25-yard touchdown pass to Cody Hoffman early in the fourth quarter.
After a wild first half that lasted nearly two hours, Houston had taken a 38-34 lead.
BYU's offense racked up 456 total yards, even with a lull that resulted in three straight punts in the middle of the half. But Houston scored via a kickoff return and an interception return, in addition to picking off a pass in the end zone.
One team or the other scored points on BYU's first five possessions. Jamaal Williams ran 15 yards for a touchdown, Hill threw two TD passes to Ross Apo and Justin Sorensen kicked a 41-yard field goal, but Houston's Derrick Mathews intercepted a Hill pass and returned it 29 yards for a score.
The craziness started with an interception by BYU's Spencer Hadley on the first offensive play of the game and continued with Demarcus Ayers' 95-yard kickoff return for a Houston touchdown.
Incredibly, BYU posted 275 total yards and 18 first downs in the first quarter to UH's 90 yards and two first downs, yet that statistical domination was good only for a 24-21 lead.
The biggest momentum-changing play of the half came with BYU leading 24-21 and threatening after a Daniel Sorensen interception. Hill threw another interception on a second-and-goal play from the UH 4-yard line.
Houston then charged ahead 31-24 before Williams ran 1 yard for a tying score with 3:57 left in the half. But then Houston got a 41-yard touchdown pass from O'Korn to Daniel Spencer to make it 38-31. Sorensen's 20-yard field goal cut Houston's lead to 38-34 on the final play of the half.