The Cougars took over at their 15 and drove to the UCLA 42, but Mangum was pressured and his desperation throw was picked off by Jack.
Trailing 20-10 after a BYU field goal early in the fourth quarter, the Bruins drove 68 yards and trimmed the deficit to 20-17 with a 19-yard strike from Rosen to Jordan Payton, who beat BYU's Micah Hannemann on the play.
But the Cougars put together a clock-eating, 16-play drive and capped it with Trevor Samson's 32-yard field goal to go up six with 5:39 remaining.
The Bruins improved to 3-0 with the win, while BYU fell to 2-1.