Shabazz Muhammad had 22 points and eight rebounds. Drew and Travis Wear added nine points each as the Bruins overcame their worst shooting game of the season to win.
UCLA (14-9, 6-4 Pac-12) was coming off a 75-71 home loss to USC a week ago.
Aziz N'Diaye had 11 points and a career-high 18 rebounds to record his 10th double-double of the season for Washington (13-10, 5-5), which has lost six of its last seven games. C.J. Wilcox scored 15 points and Abdul Gaddy added 13.
With the Bruins clinging to a 56-55 lead, Muhammad made one of two free throws. Wilcox's lay-up tied the game at 57-all with 10 seconds left.
Drew's fall-away jumper over Washington defender Shawn Kemp Jr. was his third game-winning basket of the season.