Brandon Taylor had 13 points and Dallin Bachynski and Delon Wright each scored 12 for the Utes (17-9, 6-8), who lost at home for the second time this season.
The Utes had several opportunities to win the game, but missed nine free throws in the second half and overtime.
"It was a classic game," said head coach Larry Krystkowiak. "One of the things that I was most proud of was the fact that we came back. We build that deficit against a really good basketball team and put a little bit different line up on the floor. We went small. I thought some of our young kids gave us a great lift. We got more aggressive. We got to the free throw line, only five fouls in the first half. We came back and stuck with it."
The score was tied at 58 before Hollis-Jefferson made two baskets to give the Wildcats a four-point lead with 1:52 left in overtime. His second bucket came after Taylor, an 86 percent free throw shooter, missed three straight from the line to prevent Utah from taking the lead.
Wright made a pair of free throws to cut it to 62-60, but threw the ball away with 33.9 seconds left. Johnson and T.J. McConnell each made free throws to help secure the victory in the final seconds.
Utah lost despite outrebounding Arizona 37-31 and finishing with a 34-23 edge in points in the paint.
"This wasn't the most disappointing game we’ve had but they played well - we missed some free throws and we could not close it out," said center Dallin Bachynski.
Utah kept pace by hitting seven of its first 10 shots from the field. Princeton Onwas stole the ball from Hollis-Jefferson and dunked it on the other end to cap a 6-0 spurt.
Arizona went on a 14-0 run later in the half. York capped the surge by hitting a 3-pointer and then taking his own steal from Taylor in for a layup, giving the Wildcats a 25-16 lead.
Utah briefly trimmed the deficit to five after baskets from Taylor and Bachynski. Arizona quickly answered, pushing the lead to double digits with 3s from York and Hollis-Jefferson that made it 35-24.
The Utes cut it to single digits again when Onwas took another steal in for a dunk to make it 35-26. Arizona had a chance to take a 12-point lead into the locker room when Jordin Mayes nailed a 3-pointer at the buzzer, but the officials waved it off during halftime.
Utah cut the lead to 40-33 when Taylor stole the ball from Johnson and took it in for a layup. The Wildcats thwarted the rally for a time, going back up by 12 on McConnell's jumper.
But the Utes ripped off a 10-2 run to get back into the game. A pair of big baskets from freshman Ahmad Fields cut the deficit to 48-44. Arizona endured a short cold spell after a layup from McConnell made it 52-44 with 6:41 left, going 3:13 without a field goal.
Utah took advantage with an 11-2 run and went ahead 55-54 on a runner from Jordan Loveridge with 3:09 remaining.
Arizona regained a one-point lead on Johnson's jumper with 2:21 left. Utah tied it when Loveridge made one of two free throws with 28 seconds to go. McConnell had a chance to win it for the Wildcats in regulation, but missed a runner in the lane as time expired.
"It is a pretty fine line of how this game goes," said Krystkowiak. "Even at the end of the first half, we got mental things that we have to clear up. We were trying to run a play and get the last shot of the half and for some reason we run the play with 12 seconds left and give them a shot at the end, that ended up not counting. That is a mistake that we made a week ago. There are plenty of mental things that we need to clean up but the kids battled."
Utah next hosts Arizona State Sunday night at 6:00 p.m.