Allen also had a key offensive rebound that led to Glen Davis' two foul shots that made it 87-83 with 15.2 seconds remaining. Davis also had 20 points.
The Celtics trailed 56-41 early in third quarter, but closed the period with a 23-6 run to go ahead 67-64.
David West led New Orleans with 32 points. Marco Belinelli added a season-high 23 points but Chris Paul went without a basket for only the fourth time in his career, finishing 0-of-9 from the floor.
Paul barely grazed the rim on a runner in the lane that would have tied it at 83 late in the game, and Emeka Okafor was tied up for a jump ball as he went for the putback. The Hornets controlled the tip, but Belinelli was stripped as he drove to the basket.
Kevin Garnett scored to give the Celtics an 85-81 lead with 57.4 seconds. West followed with a driving bank shot, but the Hornets couldn't get any closer.
Allen grabbed an offensive rebound off a miss from Davis seconds later. After Paul Pierce called a timeout to avoid a turnover while being trapped near midcourt, Davis was fouled and hit both free throws.
Okafor scored on a putback, but the Hornets could not prevent Allen, one of the best foul shooters in NBA history, from getting the inbounds pass.
Garnett finished with 12 points and nine rebounds. Pierce went 1-for-9 from the floor, finishing with nine points and six assists.
Okafor had 10 points and 11 rebounds for the Hornets. Trevor Ariza also had 10 points.
The Celtics' huge spurt in the third quarter started when Allen took an inbounds pass after a Boston timeout and drained a 3-pointer. Davis then scored twice in succession and Allen had three baskets, the last on a give-and-go from Garnett that put the Celtics ahead 65-64.
Davis finished the quarter with another easy layup, and Delonte West converted a three-point play at the start of the fourth to give the Celtics a 70-64 lead.
The Hornets managed just eight points in the final 10:45 of the third and two — on a West bucket — in the last six minutes.
Paul had seven assists in the first quarter, helping New Orleans to a 28-14 lead. He finished with four points and 15 assist.
Notes
The Hornets do not play again until Thursday at Utah. The layoff is their longest of the season other than the All-Star break. … Boston is 1-4 in the second game when it plays on the road for two consecutive days.