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.