Seconds earlier, Vince Carter had a potential game-tying drive bounce off the rim.
Rajon Rondo added 20 points as the Celtics shot 61.2 percent from the field in beating the Nets for the third time this season. Kendrick Perkins had 17 points and 13 rebounds and Allen finished with 16 points.
Carter had 34 points for New Jersey, which had taken a three-point lead after rallying from a 13-point deficit early in the fourth quarter. Devin Harris added 25 points and 11 assists.
The game marked the return of outcast Knicks guard Stephon Marbury to the New York metropolitan area and the former local star - and ex-New Jersey Net - was booed every timed he touched the ball by the almost 16,000 fans at the Izod Center.
It was entertaining and Marbury even managed a couple of smiles despite having a shot blocked by Sean Williams and having a 3-pointer at the shot clock hit nothing but air.
When the game was on the line, it was in the hands of Pierce, who hit 12 of 14 from the field.
With the game tied at 111, the Celtics had trouble getting off a play and coach Doc Rivers called a timeout with 25.5 seconds to go and 6 seconds left on the shot clock.
The ball was eventually inbounded and quickly given to Pierce on the left wing. He looked at the shot clock, started a move to the basket and then pulled up for a sweet jumper that ripped the net.
Carter, who was 13 of 21 from the field, got the call for the Nets after their timeout. He got the ball at the top of the key, drove down the right side of the lane and powered his way for a layup as the Celtics challenged him. His shot bounced off the rim and Perkins got the rebound and was fouled.
With the Nets not over the foul limit, the Celtics had to inbound the ball again and Allen nailed down the win for Boston.
Rondo scored 16 points in the third quarter in helping the Celtics take an 11-point lead. The margin grew to 13 when Mikki Moore hit a basket early.
Down by 11 with 5:37 to play, the Nets ran off a 19-5 spurt that featured a 3-pointer and a three-point play by Harris, a 3-pointer and three free throws by Jarvis Hayes, who added 16 off the bench for New Jersey.
The Nets eventually took a 111-108 lead on a 3-pointer by Carter with 1:14 to play.
The Celtics called a time out and ran a screen on the left wing, setting up Allen for a game-tying 3-pointer.
Harris missed a floater with 45.6 seconds to go and that set the stage for Pierce.
Notes: Celtics star Kevin Garnett missed his sixth straight game with a strained right knee. Boston is (4-2) without him. ...Allen was 7-for-7 from the free throw line ...Rivers wasn't happy with the officiating in the first half - 17 personal fouls against Boston - and got a technical for complaining to Bennett Salvatore about it. Marbury finished with no points on 0-for-2 from the field with four assists. The Celtics are now 19-0 when Rondo scores at least 18.
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