In an early Pac-12 Conference showdown, the Wildcats remained one of five unbeaten teams in the country.
The Bruins trailed by 12 near the midway point of the second half, but came back to take a 70-69 lead with 1:44 to go on two free throws by Bryce Alford.
Nick Johnson then put Arizona back in front for good with a layup and the Wildcats, with the game on the line, made 8-of-10 from the foul line the rest of the way.
York finished it off with two free throws that came with 13 seconds remaining and gave Arizona a four-point lead.
Johnson led Arizona with 23 points and Kaleb Tarczewski had 16 points and seven rebounds.
At 16-0, Arizona has equaled the best start in school history. The Wildcats are 3-0 in Pac-12 play and are part of an unbeaten group that includes Syracuse, Iowa State, Wisconsin and Wichita State.
Kyle Anderson scored 16 points and had 11 rebounds for UCLA (12-3, 1-1 Pac-12). The Bruins' other losses have come on the road against Missouri and Duke.