The Cougars have now won seven straight over Utah, and 11 of the last 12.
Tyler Haws' three-point play with 2:48 remaining gave BYU a 59-58 lead and the sophomore returned missionary made a pair of free throws with 16.7 seconds left to seal it.
Utah had several chances in the last 16 seconds to tie it, but Glen Dean fired up an airball and Justin Seymour missed a desperation 3-pointer at the buzzer after a timeout with four seconds left.
The Utes had a 35-26 halftime lead, and led by as many as 12 points in the first half.
Matt Carlino led BYU with 19 points off the bench, and hit several huge 3-pointers in the second half when the Cougars were making their comeback. Jordan Loveridge led Utah with 14 points, and Dean added 10.
The Utes shot 46 percent in the first half while holding the Cougars to 29 percent to get the big lead.
The Cougars took their only lead of the first half when Carlino came off the bench and hit a pull-up jumper to make it 7-6.
But Dean's 3-pointer got the Utes going again, and Utah didn't trail the rest of the first half.
A forced 3-pointer by Renan Lenz gave the Utes a 20-15 lead with 8:10 remaining, and when Dean made another 3-pointer four minutes later, the Utes had a 27-17 lead and the crowd was stunned.
The Cougars made a mini-run in the final few minutes of the half, but Jason Washburn's three-point play restored Utah's momentum and sent the Utes into the locker room with a nine-point lead.