Notre Dame's streak of not giving up an offensive touchdown ended just past the midway point of the second quarter as BYU took a 14-7 halftime lead.
The Irish, who had gone four games without giving up a touchdown, gave up two TDs in 2:18 in the second quarter.
The Irish have been plagued by mistakes in what was not supposed to be a trap game. Notre Dame has lost 36 yards on 16 penalties, and kicker Kyle Brindza missed two field goals in the opening half.
Quarterback Tommy Rees, who started for the Irish in place of Everett Golson (concussion), was 0-for-6 passing in the second quarter with passes sailing high and one going through a receiver's hands for an interception. He went 6-for-7 passing in the first quarter with a touchdown throw to tight end Tyler Eifert, a Bishop Dwenger grad.
BYU made Notre Dame pay for a bad penalty and blown coverage when it tied the game at 7 with 8:25 left in the first half.
Notre Dame safety Matthias Farley was called for a late hit after BYU running back Jamaal Williams had gained 21 yards to put the ball on the Irish's 6-yard line.
Three plays after the penalty, quarterback Riley Nelson found receiver Cody Hoffman all by himself in the middle of the end zone for a 6-yard touchdown catch as BYU tied the game at 7 with 8:25 left in the first half.
Hoffman's score was the first offensive touchdown Notre Dame had given up in 17 quarters, and it was the first TD the Irish gave up in the red zone in opponents' last 12 trips inside their 20.
Notre Dame gave up another TD just two minutes later after receiver DaVaris Daniels had a pass go through his hands, bounce off his helmet and into the arms of BYU linebacker Kyle Van Noy.
The Cougars, who started with the ball on the Irish's 30, got helped out by a 10-yard penalty for holding by Notre Dame and scored another TD with Nelson hitting Kaneakua Friel for a 2-yard scoring pass with 6:07 to play in the opening half.
Notre Dame used Tommy Rees and Andrew Hendrix at quarterback on it next drive, but Irish kicker Kyle Brindza missed his second field goal of the