Brandon Taylor made a three-pointer with 1:57 remaining to give Utah a 57-55 lead. York then drew Jakob Poeltl's fifth foul and made one free throw and then rebounded his miss on the second attempt and put it in to give the Wildcats a 58-57 lead with 1:39 left.
Hollis-Jefferson, who had 10 points, made three of four free throws in the final minute and York hit two more to clinch the Wildcats' 13th consecutive win over Utah (22-6, 12-4 Pac-12).
Kaleb Taczewski led Arizona (26-3, 14-2) with 13 points while Stanley Johnson had 12 points and 11 rebounds. The Wildcats now lead the conference standings by two games with two games remaining.
After dominating Utah, 69-51, earlier in the season thanks to a big rebounding advantage, the Wildcats again bullied the Utes on the boards, 44-35, to overcome 33% shooting.
Delon Wright had 17 points and Jordan Loveridge scored 10 before fouling out as the Utes finished the season 16-1 at home.
Taylor, Utah's second leading scorer, got his first points of the game on a three-pointer to put the Utes up 47-41, with 8:11 left and get the sellout crowd roaring.
But York hit a 3 and Hollis-Jefferson dunked to draw within one point again. It was back-and-forth from there as the game had seven lead changes in the last 7 minutes.
Under considerable harassment from the Utes, the Wildcats missed their first 10 field goal attempts of the second half before Johnson stole the ball and made a layup.
Hours after the Utes beat Arizona State, 83-41, on Thursday, Utah students began erecting a tent village around the Huntsman Center in efforts to get the best seats.
The buzz for a college basketball game in Salt Lake City hadn't been this intense since the 1996 New Years Eve matchup between No. 7 Utah and No. 2 Wake Forest, when future No. 1 NBA draft pick Tim Duncan scored 23 points and future No. 2 pick Keith Van Horn tallied 24 in the Deacons' 70-59 win.
The first half was the physical battle many expected between the conference's two best defensive team. Arizona jumped out to a 17-8 lead but the Utes fought back and took a short lead before the half ended with Arizona ahead 29-28. The teams were whistled for a combined 25 fouls.