Joseph Young added 14 points and Elgin Cook had 12 for the Ducks (20-8, 10-5 Pac-12). They improved their NCAA tournament chances with their first victory over a top 10 team since beating then-No. 3 Arizona last year in the regular-season finale.
Delon Wright scored 20 points for Utah (21-5, 11-3). The Utes dropped a game behind Arizona in the conference race with a showdown against the No. 7 Wildcats set for Saturday in Salt Lake City.
Brooks, a freshman, scored seven consecutive points to put the Ducks up by 10 with 4:58 to play. After Utah had a 7-0 run to pull within three again, Brooks added four more points — the last two on a driving dunk — to give the Ducks a 59-52 cushion.
Oregon sealed the win by hitting seven of eight free throws in the final 1:09. The Ducks were 15 for 18 overall from the foul line.
Utah made just 10 of 31 shots (32.3%) in the second half and finished eight for 29 (27.6%) from three-point range after coming in as the Pac-12's best beyond the arc at 40.6%.
The taller Utes, who also rank second in the Pac-12 in rebounding margin, saw the Ducks grab a 23-14 edge on the glass in the second half. Oregon finished with a 35-31 advantage in rebounds.
Utah jumped to a 7-2 lead but lost 7-foot freshman Jakob Poeltl to his second foul after just 4 1/2 minutes were gone on a play that also wiped out a three by Wright.
Poeltl, the Utes' top rebounder, sat out the rest of the half and finished with five points and four rebounds.
Both teams struggled on offense until Young's layup gave Oregon its first lead at 21-19 with 5:30 left in the half. Young's first points also started a 9-2 run that helped lift the Ducks to a 32-27 halftime advantage.
Oregon forced the Utes into 11 turnovers in the first half and converted those into 11 points.
Washington 87, at Washington State 84: Andrew Andrews scored a career-high 35 points, including a game-winning three-pointer with 2.7 seconds left, to lift Washington the Huskies to an a triumph over the Cougars.
Nigel Williams-Goss added 20 points and six assists for Washington (15-11, 4-10), which snapped a seven-game losing streak.
DaVonte Lacy led Washington State (11-15, 5-9) with 23 points. Josh Hawkinson had 17 points and 14 rebounds, Que Johnson scored a season-high 16 points off the bench and Jordan Railey contributed 10 points.
The Cougars led, 41-37, at the half. No more than three points separated the teams most of the second half.