It was only Boston's second victory in seven games, The Associated Press reported.
"I thought we were gassed at the end of the game," Celtics coach Brad Stevens said. "We missed some defensive assignments, and I thought we missed some good opportunities on offense, but they found a way."
The Celtics had struggled in tight games over the last two weeks, with three losses in games decided by four points or fewer, including most recently a 109-105 loss to Golden State on Saturday. Denver nearly added to their misery.
After Irving's miss with seven seconds to go, the Nuggets – not using a timeout – raced up the court and got a look at the buzzer, but Torrey Craig's tip-in of Will Barton's long attempt from three-point range was too late.
Afterward, Denver coach Michael Malone defended his decision not to use a timeout to set something up.
"(We) got a stop, didn't want them to set their defense," Malone said. "Will Barton has made plays like that before for us and I stand by that decision."
Irving finished 11 of 17 from the field to lead the Celtics. Jayson Tatum added 20 points and Marcus Morris had 14 off the bench.
Boston blew a 20-point lead and trailed heading into the final period. Then the Celtics couldn't hold onto a six-point advantage in the final two minutes as the Nuggets tied the game at 108 with 43 seconds to go.
Moments later, though, Al Horford fed Brown for the decisive three. The second-year small forward had missed his first four three-point attempts.
"I was like, no way I'm missing this one," Brown said. "I was open. Everything that happened before didn't matter. I had to make the shot."
Nikola Jokic had a double-double by the first minute of the second half for Denver, finishing with 24 points and 11 rebounds. Barton scored 19, Gary Harris 15 and Jamal Murray 14.
Mason Plumlee had 16 points and eight rebounds before leaving with a strained right calf in the fourth.
"We could have beat a really good and tough team, but we didn't," Jokic said. "We had a chance, but we didn't take it."
The Celtics beat Denver for the second time this season, also prevailing 124-118 on December 13.
In this one, Boston made 10 of its first 16 shots from beyond the arc to go up 51-31 in the second quarter and led 56-45 at halftime. Irving paved the way with 15 points on 6-of-7 shooting.
The Nuggets climbed back on the strength of a 14-5 run to end the half and outscored Boston 37-24 in the third to go up 82-80.