Cobbs capped a big second half with his game-winning jump shot over Kaleb Tarczewski, which came after Nick Johnson missed a try in the lane for the Wildcats.
The senior point guard scored all but three of his 19 points in the second half as the Golden Bears (15-7, 6-3 Pac-12) defeated a top-ranked team for the first time since Jan. 30, 1994.
David Kravish added 14 points and 11 rebounds for Cal, which improved to just 3-24 all-time record against No. 1 teams.
Tarczewski supplied 18 points for the Wildcats (21-1, 8-1), who were knocked from the ranks of the unbeaten after losing forward Brandon Ashley to an injury in the first half.
With Arizona's loss, No. 2 and still-unbeaten Syracuse is likely to take over the nation's top spot.