Jayson Tatum scored 37 points to lead Boston. Dennis Schroder added 26 points while Al Horford chipped in 21 points and nine assists.
Robert Williams III scored on three straight possessions to cap a 9-0 run and give Boston a 113-109 lead midway through the fourth quarter. Utah answered by scoring baskets on seven straight possessions to surge ahead 127-120 with 1:57 left. Mitchell made three shots in a row, including a go-ahead three-pointer.
Tatum made a layup and two free throws to get Boston within three in the final minute. Mitchell hit a stepback three-pointer with 28.2 seconds left to stave off a comeback.
Conley made three three-pointers to fuel a game-opening 14-2 run for the Jazz. It set the stage for a first quarter when Utah made nine three-pointers and shot 12-of-18 from the field overall.
Boston trimmed the deficit to 30-23 on a steal and dunk from Josh Richardson. Jordan Clarkson answered with back-to-back baskets to boost Utah's lead back to double digits.
The Jazz led 43-29 in the second quarter on a dunk from Gobert. Then, the Celtics rallied. Boston ripped off an 11-0 run punctuated by back-to-back baskets from Tatum and Schroder to cut Utah's lead to 43-40.
Horford scored five baskets in the second quarter to keep the Jazz from pulling away again before halftime. His final one, a driving hook shot, pulled the Celtics within 57-56.
After trailing the entire first half, Boston took its first lead at 73-71 in the third quarter on a reverse layup from Marcus Smart.
Celtics: Jaylen Brown did not play (right hamstring tightness). ... Smart had four steals, giving him 750 for his career.
Jazz: Royce O'Neale returned to action after missing two games with a right foot sprain. O'Neale finished with 12 points, 10 rebounds, and five assists. ... Utah scored a season-high 38 points in the first quarter and a season-high 41 in the third quarter.
Celtics visit Portland on Saturday.
Jazz visit Cleveland on Sunday.