The Golden Bears rallied from a first-half slump and held Oregon to 27 percent shooting. Oregon slipped back into a tie with Arizona for the lead in the Pacific-12 Conference race, a half game in front of UCLA.
Oregon's Damyean Dotson missed a jumper in a tie game with 36 seconds to play. California then ran the clock down before Cobbs put up his game-winning jumper from the left side.
Cobbs had 14 points, eight rebounds and seven assists.
California (17-9, 9-5 Pac-12) scored the game's first six points before Oregon went on a 16-2 run. The Ducks, however, finished with just 16 field goals and they went 2-of-15 from 3-point range.
Oregon (21-6, 10-4), which had won three in a row, was led by Carlos Emory with 13 points.