Brown, the Pac-12's scoring leader at 21.7 points per game, shot 9 of 19 from the field and added four rebounds and five assists. Bey went 5 of 12 from the field including 3 of 6 from 3-point range as the sixth-seeded Huskies (17-14) finished 8 of 18 from outside the arc.
Washington advances to face third-seeded Southern California (25-6) in Thursday's quarterfinals. The 21st-ranked Trojans beat the Huskies 79-69 in Los Angeles on Feb. 17 during the teams' lone regular-season meeting.
Eleventh-seed Utah (11-20) couldn't overcome 16 turnovers, which Washington converted into 21 points. The Utes forced only five turnovers that produced just two points.
Cole Anthony Jr. had 18 points, Branden Carlson scored 14 and Gabe Madsen added 12 for the Utes, who finished with four consecutive losses.
Washington led 40-37 at the half, but two turnovers enabled Utah to tie it at 44 on a Carlson putback with 16:56 left. The Utes couldn't get in front, though, and Brown soon scored three consecutive baskets for the Huskies, the last starting a 12-0 run that built a 63-48 lead with 9:30 remaining.
Utah cut the deficit to 10 on multiple occasions, including with 4:27 to go on a Carlson dunk, but Bey thwarted a rally with a three-point play to put Washington up 74-61 with 3:59 remaining.
The first half saw six ties and six lead changes, with Washington building a six-point edge on a Brown jumper with 2:59 to go. Utah shot 50 percent in the first half but gave up 17 points off turnovers or second-chance shots.
Cole Bajema had 11 of his 16 points for Washington in the first half, and he ended the night 3 of 4 from 3-point range.
Washington earned its first victory away from home since Feb. 3, and the Huskies ran their overall win streak to three games.
Utah recorded its first 20-loss season since going 6-25 in 2011-12, its first year in the Pac-12.