Washington shut out the Cougars in the second half to complete a 9-4 season. The Huskies won a bowl game for the first time in three tries.
Quarterback Taysom Hill scored on a 1-yard run for BYU early in the second quarter to tie the game at 7-7, but Qualls ran the ensuing kickoff back all the way.
The Cougars closed within a point late in the first half on two field goals by Justin Sorensen, after which Washington expanded its lead on the second touchdown run of the game from Bishop Sankey.
Washington owned a 21-16 halftime lead, then put the game away on a third-quarter touchdown throw of 16 yards from Keith Price to Austin Seferian-Jenkins and a fourth-quarter field goal from Travis Coons.
Washington's win gives the Pac-12 Conference a 3-1 bowl record this season with five more teams from that league left to compete.
Price completed 18-of-32 passes for 128 yards with a touchdown and an interception. Sankey carried 21 times for 95 yards.
BYU wound up at 8-5 this season. The Huskies overcame 426 total yards produced by Hill -- 293 through the air and 133 on the ground.