Rudy Gay's short pull-up jump shot for Memphis with 7.3 seconds left in regulation tied it at 99.
Celtics forward Glen Davis hit a jump shot and Allen dropped in an arching 3-pointer to give the Celtics a 104-99 advantage in overtime.
Pierce made four free throws in the final 1:30 to seal Boston's seventh win in its last eight games. The Celtics finished their four-game road trip 3-1.
Gay had 22 points for the Grizzlies, who had trailed by as much as 11 points. Memphis forward Zach Randolph scored 21 points and grabbed 11 rebounds.
Celtics center Shaquille O'Neal scored 18 points in his second game back after missing five with a bruised knee. Rajon Rondo had 11 points and 17 assists.
Allen scored 15 points after going for a season high 35 in a 112-107 win at Miami on Thursday.
Memphis grabbed its first lead of the game when a short shot by Gay made it 86-85 with 6:44 left in the game.
Kevin Garnett's jumper gave the Celtics the lead again, but the Grizzlies took it back on a layup by rookie guard Greivis Vasquez, making the score 88-87 about halfway through the fourth.
A shot by Gay gave the Grizzlies a three point lead, but the Celtics went on an 8-2 run to take a 97-94 lead on a Davis layup with 3 minutes left.
Allen's jumper made it 99-95 before Gay scored two baskets in the final 39 seconds to force overtime.
Boston is 3-0 in overtime this season, while Memphis is 0-2. The Celtics now have five consecutive wins on Memphis' home court.
Grizzlies owner Michael Heisley honored Gay before the game with a Rolex watch for his contribution to the U.S. men's national team's gold medal performance at the 2010 FIBA World Championships. Gay averaged 7 points and 2.9 rebounds in nine games in Turkey.
Notes: Boston's roster has 109 seasons of combined NBA experience. Memphis' has 36. … Celtics center Jermaine O'Neal (sore left knee) was not with the team … Conley and Rondo are two of only three NBA players this season with more than 60 assists and 20 steals. Washington rookie guard John Wall is the third … The Grizzlies' last home win against the Celtics was a 93-76 victory on March 14, 2006.