"It almost came down to that we had Paul and they didn't have Wade," Boston coach Doc Rivers said.
Wade missed the game with a right hip flexor, and the Heat missed his NBA-best 29.9-point scoring average.
"We're a team, the Miami Heat, with or without D-Wade," said Michael Beasley, who led Miami with 21 points off the bench. "Obviously, he's a big part of our team and a majority of the scoring. But when he's out, when he's not playing, somebody's got to step up and a couple of guys did."
The Celtics could use Garnett's presence after he sat out his 13th straight game with a strained right knee.
"We miss his leadership out on the floor, his verbal leadership," Rivers said. "He's the one guy that holds everybody accountable defensively."
Leon Powe, Tony Allen and Brian Scalabrine also are sidelined, and Allen hyperextended his elbow Tuesday night in a loss at Chicago. Glen Davis started at forward in place of Powe after missing the previous four games with a sprained right ankle.
The Celtics hope to get all their players back for the playoffs and a run at defending their NBA championship.
The division title "really doesn't mean anything to the Boston Celtics," Pierce said. "They don't put that banner up. Maybe in other arenas they put that banner up, but here it really doesn't mean a thing."
Boston improved to 51-18. Second-place Philadelphia can win no more than 51 and the Celtics have won the season series with the 76ers.
The Heat did not say how long Wade would be sidelined, but he was not ruled out of games Friday night at New Jersey and Sunday against Detroit.
The close game "showed that even without D-Wade we're still going to fight," Chalmers said. "We're a good team."
With the score tied at 100 after regulation, Eddie House and Pierce hit 3-pointers and Rondo sank a layup for a 108-100 lead as Miami went cold from the field.
Then Chalmers made a 3-pointer and, after Kendrick Perkins hit a free throw, Chalmers sank another 3-pointer, making it 109-106 with 44 seconds left.
"Mario did a fantastic job," Miami coach Erik Spoelstra said, "basically doing the things that Dwyane normally does."
But House hit one free throw and Pierce sank two before Beasley's basket ended the scoring. Boston went 3-for-6 in overtime while Miami was just 3-or-13.
"We know that we are injured," Pierce said after Boston's second win in six games, "but just to get back on the winning mind-frame is big for us."
With all their injuries, the Celtics relied on Pierce even more than they usually do.
Pierce went 14-for-23 from the field and had 12 points in the first quarter,
"He was scoring on every one of our matchups," Spoelstra said.
In its previous four games, Miami beat Chicago in double overtime, Boston in regulation and Utah in triple overtime before losing at Philadelphia on Sunday.
Boston had a chance to win in regulation Wednesday, but Pierce's 18-footer from the left missed as the buzzer sounded.
The lead changed hands 11 times in the fourth quarter when the biggest edge was just three points by Miami.
Boston led 34-21 after one quarter, but Miami opened the second period with a 10-2 run that cut the margin to 36-31. The Celtics stretched that to 45-37 before the Heat ended the half with a 17-4 spurt and a 54-49 lead.
Daequan Cook, starting in place of Wade, gave Miami its first lead, 46-45, on a 3-pointer.
Boston big men Davis and Mikki Moore each picked up his fourth foul in the final minute of the first half. Both fouled out in the fourth quarter. ... Cook made his second start of the season. ... Pierce had 11 rebounds and Rondo added 10 assists. ... Miami outscored Boston off the bench 46-24.