Shaquille O'Neal had 13 points and nine rebounds, but he missed two free throws with 1:02 left and Boston nursing a two-point lead. Philadelphia had two chances to tie it, but Andre Iguodala slipped and turned the ball over the first time and then had his shot blocked by Kevin Garnett the next.
Allen hit a pair of free throws with 5.6 seconds left to ice it. Elton Brand had 16 points and 12 rebounds, and Jrue Holiday scored 15 points for Philadelphia, which was coming off a 45-point loss to the Chicago Bulls - its worst in three years.
The Celtics have not lost since Nov. 21, the second game of their only two-game losing streak of the season. Their 14-game winning streak matches the fifth-longest in the history of the NBA's most-decorated franchise, and it ties the third-longest in 46 years.
The 2007-08 team, which won Boston's 17th NBA title, won a franchise-record 19 in a row.
Lou Williams scored 12 points and ex-Celtic Tony Battie had 10 for Philadelphia, which lost 121-76 to the Bulls on Tuesday night. That had Celtics coach Doc Rivers sending angry text messages to his former assistant, Chicago's Tom Thibodeaux.
"You don't want to be the next one, but we are," Rivers said before the game.
The Celtics opened a quick lead in the first and led 23-17 by the end of one, but the Sixers scored seven straight points, five of them by Williams, to take the lead early in the second. It was 44-38 at halftime, and Pierce hadn't scored; Garnett had just four points and a team-high three rebounds.
Brand made a pair of jumpers early in the third that helped Philadelphia hold onto an eight-point lead, but after falling behind 48-40 the Celtics scored 14 of the next 16 points. Pierce, who missed his first seven shots in the game, hit a jumper with 8:42 left in the third and then followed it with a 3-pointer from the left side that cut the deficit to one.
But he did not make another basket until he stole the ball from Jodie Meeks with 5:20 left in the game and went in for a slam dunk that cut a four-point deficit to two points, 72-70. He made another, a fallaway jumper from just outside the key, with 96 seconds left to give Boston an 82-78 lead.
Brand made a short jumper, then fouled out when he pulled O'Neal down to keep him from scoring an easy layup. O'Neal missed both foul shots, but Iguodala slipped near the lane and Garnett came away with the ball.
After Boston failed to score, Iguodala went to the basket but Garnett tipped the shot and Pierce came away with the rebound.
NOTES: Celtics coach Doc Rivers said he expects Jermaine O'Neal to be ready for Saturday's Christmas Day game in Orlando. O'Neal has not played since Nov. 8 because of a sore left knee, and Rivers said he came down with the flu just when he was ready to return. ... Celtics point guard Nate Robinson left the game late in the first quarter to get his ankle re-taped, the team said. In the second half, he had ice on his right thigh. ... Red Sox designated hitter David Ortiz helped the Celtics mascot, Lucky, with an acrobatic dunk before the third quarter. But the stunt had to wait so Ortiz could record it on his cellphone. ... The Celtics were called for two technical fouls in the third quarter, on Garnett and Pierce, for complaining about calls. ... Sixers guard Holiday had nine points in the first quarter but did not score again until the fourth.