Thirty-five-year-old Paul Pierce led Boston with 26 points, including seven straight in the extra session.
"It was huge," Celtics coach Doc Rivers said. "Whatever that was, it was the best win of the year for me. . To go into overtime and still have enough to win."
It wasn't just the ageless Pierce. Kevin Garnett, three months shy of his 37th birthday, stood strong at the end, refusing to let Rivers sub him out by insisting, "I am good."
Rivers didn't believe that, "but I kept him in and he was terrific."
Garnett had four points in overtime on 2-of-2 shooting, with three rebounds. He finished with 13 points, 10 rebounds, a blocked shot and steal.
The Celtics also got a big game from Avery Bradley, who scored a season-high 18 points on 8-of-12 shooting.
Pierce had a chance to win it in regulation, but his 19-footer at the buzzer rimmed out.
Alec Burks' reverse layup pulled Utah to 108-105 with 37 seconds left in overtime. Garnett's banked 3-pointer with 13 seconds left came after the shot clock expired, giving the Jazz another chance.
Paul Millsap was fouled before he could get off a 3, but made two free throws with 4.2 seconds remaining.
Courtney Lee added two free throws at the other end with 1.2 seconds left to bump Boston's lead back to three, and Randy Foye's 26-footer at the buzzer missed everything.
It was another one the Jazz let get away, though Monday's game had huge swings both ways from start to finish.
Overall, the game had 13 lead changes and was tied 17 times.
"We were in position," said Jazz forward Marvin Williams. "We fought back in the fourth quarter to force overtime but Boston just made big plays down the stretch."
Gordon Hayward led Utah with 26 points, Millsap had 16 and Al Jefferson finished with 15 points and 11 rebounds.
The Celtics (30-27) were playing their fifth game in seven days, but didn't seem to care down the stretch.
The Jazz (31-26) led 101-99 in overtime on DeMarre Carroll's 21-footer, but Pierce countered with a 3-pointer, then followed with a pull-up jumper over Carroll and a 15-footer to give the Celtics a 106-101 edge with 1:12 left.
Jefferson hit a 15-footer with 1:05 left to get Utah within 106-103, but Garnett's jumper helped seal it.
"We knew this was probably going to be the toughest game for us physically and mentally," Pierce said. "Talking about a long road trip, coming in to one of the toughest places to play. We felt we could salvage this trip with a win here. So guys did a good job of just being mentally tough, digging in and doing what we had to do to get the win."
The Celtics, who went 2-3 on the trip, also were smart down the stretch, fouling with fouls to give and finding a wide-open Lee on the inbounds pass when the Jazz needed a late steal in overtime. His free throws provided the final margin.
The Jazz trailed by eight entering the fourth but opened on a 13-2 run.
Jefferson's 10-foot turnaround jumper over Brandon Bass tied it at 93 with 2:46 left in regulation.
Pierce hit an 18-footer with Carroll diving at him for a 97-95 Boston lead with 36 seconds remaining in regulation, only to see Burks tie it with a tough layup with 19 seconds left.
"We wanted to win in regulation," Pierce said. "It would have felt better just to get a stop when we needed it. That's what we need to get better at. We didn't do it in Portland. We didn't do it tonight. That's what we need to start focusing on. When we get the lead and we need crucial stops, we have to figure out how to get them."
The game took a 16-point swing in the third, as Boston trailed 58-50 only to counter with a 20-4 run and lead 80-72 entering the fourth.
Pierce ignited the run with a 3-pointer, Bradley hit two more 3s and Lee added a dunk after a steal and another 3-pointer. Pierce capped the run with a jumper over Hayward for a 70-62 Boston lead.
The Celtics hit 6 of 13 3-pointers in the 32-point third quarter, while Utah made just 5 of 16 from the field.
It was the same story as Saturday, when the Jazz fought back early only to see the Los Angeles Clippers go on a 23-4 run and douse any hopes.
A Jazz team that had won three straight and seven of 10 has now dropped two in a row.
While Hayward showed he is recovered from a right shoulder injury despite missing a pair of shots in overtime, the Jazz still need point guard Mo Williams back from a thumb injury.
Monday, Earl Watson started over Jamaal Tinsley, but Burks ended up playing the position during Utah's big second-quarter run.
Pierce said Boston reverted to various defenses to slow the Jazz.
"We did a lot of zone, did a lot of man and tried to force turnovers," Pierce said. "That's what we have to do. A lot of times we had to go to small because they had plenty of size and they rebound well. We just junked up the game a little bit by changing our defense."
It may have been junk, but it was a win nonetheless.
"We have champions," said guard Jason Terry, who made four 3-pointers and finished with 14 points and two assists off the bench for Boston.
"When you have champions that have been through so many tough games as we have, then you know you're in good hands. For us, we hang our hat on executing down the stretch."
NOTES; Jazz F Derrick Favors picked up his third foul with 9:06 left in the second quarter. . Jazz G Foye needed four 3-pointers to tie Mehmet Okur (129, 2006-07) for the franchise single-season record. Foye went 0-5 Saturday but hit his first Monday and finished 2 of 6. . Bradley started 5 of 5 and had 10 points in six minutes for Boston, while Millsap started 4 of 4 for Utah. . The Jazz led 53-48 at halftime.