Cal led by as many as 15 points midway through the second half, but the Utes pulled within 58-55 on Jason Washburn's 3-pointer with 55 seconds remaining.
David Kravish missed at the other end for the Golden Bears (11-7, 3-3 Pac-12) but poked the rebound out to Crabbe, who was fouled and sank two free throws with 17 seconds left.
Freshman Brandon Taylor missed a 3-pointer with 11 seconds left for the Utes (9-10, 1-6), and Justin Cobb grabbed the rebound and sank a pair of free throws to seal the win for Cal.
Cal used a 13-0 run midway through the first half to take a commanding lead.
Utah shot better from the floor than Cal did but committed 17 turnovers, including six by Jordan Loveridge.