The Golden Bears climbed within three points of the lead less than two minutes into the second half, but then managed only two points over the next six minutes.
Aboya began the run with a free throw, and then claimed a rebound and assisted on Josh Shipp's three-pointer. Aboya followed that with a layup and six UCLA players eventually scored during the game-breaking surge.
The Bruins (16-4, 6-2 Pac-10) remained in a deadlock with Washington for the top spot in the conference race. California (16-5, 5-3) fell a game behind in a third-place tie with Arizona State, which lost Thursday to Washington State.
Darren Collison scored 18 to lead UCLA, which had lost two of its previous three outings following a 10-game winning streak. Jrue Holiday added 13, Shipp had 11 and Aboya 10.
Theo Robertson scored 21 for California.