The loss spoiled the home debut of new Jazz players Devin Harris and Derrick Favors, who were acquired last week in the trade that sent for All-Star Deron Williams to New Jersey.
Al Jefferson had 28 points and a season-high 19 rebounds for Utah, which has dropped six straight at home — the team's worst home skid since 1982.
Allen hit a 23-footer for a 103-99 Celtics lead with 42.3 seconds left. Jefferson had a chance to cut it to one point but couldn't convert a three-point play.
Rajon Rondo then hit a 15-foot jumper to seal the win for Boston.
Harris finished with 19 points for the Jazz, who have lost 16 of 21, and Favors added nine points, four rebounds and a block in 15 minutes.
The win gave the Celtics a series sweep after they routed the Jazz 110-86 on Jan. 21, shooting 55.2 percent in the process. In that game, Williams had only five points for Utah.
The fans welcomed Harris with open arms, chasing after him on Sunday as he visited a local outdoor mall and giving him the loudest ovation of any Jazz player during pre-game introductions Monday.
The home crowd also got their first close look at rookie Favors.
"Hey, D-Will, thanks for the Favors," read one sign in the arena.
Though the Jazz would erase a 13-point first-half deficit, they couldn't hold off the Celtics down the stretch.
The Jazz, who are 1-2 since the Williams trade, jumped out to a 10-2 lead on 4-of-5 shooting. But the Celtics answered with a 12-0 run to take 14-10 lead and followed with a 13-0 run and led 30-21 after the first quarter.
Allen started the Celtics on their roll by hitting 3 of 3 from beyond the arc in the first quarter, while the Jazz shot just 36.8 percent as a team.
The Jazz fought back. Down 42-31 after Pierce's 3-pointer, Utah outscored the Celtics 15-4, capped by Harris' fast-break layup to tie the score at 46 with 3:19 left in the half.
Andrei Kirilenko's spinning layup started Utah's second-quarter run, and Paul Millsap added a pair of short jumpers. But it was defense that helped Utah get back in the game in the second quarter. The Jazz forced the Celtics into a pair of three-second violations, and held Boston to 38.8 percent shooting.
Jefferson had a double-double by halftime (18 points, 10 rebounds). He entered the game averaging 23.4 points, 9.2 rebounds, 2.0 blocks and shooting 55 percent in February.
Notes
Jazz reserve C Mehmet Okur is to meet with doctors Tuesday to determine if he can begin working out with the team again following a three-week break because of a lower-back problem. Even if he is cleared, Okur still would start slowly with shooting and running. … Celtics reserve G Delonte West was inactive with a sprained ankle and likely won't return until Friday. … Asked how high Favors was on Utah's draft board last year, Jazz general manager Kevin O'Connor initially said no comment before adding, "But he was higher than [No.] 3." … Harris acknowledged that it's been a while since he's had such home-crowd support. On Sunday, his 28th birthday, he stopped at Gateway Mall in downtown Salt Lake City. "I had one guy actually park his car in the middle of the road and chase me down," Harris said. "I was a little freaked out but it's a good freaked out. It's exciting."