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Chalmers had 22 of his 29 points after the third quarter for the Heat, including five of Miami's seven points in the third overtime.
"Coach said I had to be aggressive," Chalmers said.
Wade missed his second successive game with a sore left foot and James was held out after he turned his left ankle late in the third quarter Wednesday night during a 118-83 victory against Indiana. Neither star was in uniform.
Bosh hit a last-second, tying 3-pointer at the end of regulation to force the first overtime.
"It was a lucky shot," Bosh said.
Heat coach Erik Spoelstra said his team survived a test of its character.
"That game, the way it went down in the end, was an exercise in absolute endurance and mental and physical toughness," Spoelstra said.
"Both teams played their hearts out there at the end."
Joe Johnson had 20 points for the Hawks, who did not score in the third extra period. It was only the 10th time in NBA history a team did not score in an overtime period.
Josh Smith added 17 points and 13 rebounds for Atlanta, which was seeking its second victory against the Heat in four days but missed 15 of 46 free throws.
Hawks coach Larry Drew said his players ignored his warning against a letdown if James and Wade didn't play.
"There was a total, total mental letdown," Drew said, adding his players lacked the spark they had when they won 100-92 in Miami on Monday.
"I didn't see the sense of urgency I saw in Miami," Drew said. "The tendency is to let your guard down, and we certainly did tonight."
Bosh, the only healthy member of Miami's Big Three, also had 14 rebounds. Rookie Terrel Harris had nine points and 14 rebounds.
The Hawks led 93-90 when Bosh launched the tying 3-point attempt from the wing over Marvin Williams with only six-tenths of a second remaining in regulation.
Miami took a 67-54 lead on a 3-pointer by Chalmers midway through the third before the Hawks closed the quarter with a 16-1 run to take control.
Little-used rookie Ivan Johnson was a big surprise with 13 points for Atlanta. Johnson, an undrafted player from Cal State-San Bernardino, had a steal and jam to cut Miami's lead to one point and then closed the third quarter with two free throws to give the Hawks their first lead of the second half.
Spoelstra said before the game Wade "was a little bit more unlikely" than James to play. Afterward, Spoelstra said he was trying to protect Wade from a more serious injury.
"He's making progress with it, but we really want to take care of this before it's something that can linger," Spoelstra said.
Wade hurt his foot at Charlotte on Dec. 28.
James participated in pregame warmups but his status for Miami's game at New Jersey on Saturday remains uncertain.
"He's walking around and hopefully he'll be able to play soon," Spoelstra said. "Whether that will be Saturday, I don't know."
James and Wade were often active and animated as they watched from Miami's bench, at times leaping up as they reacted to plays.
"I hope the guys realize it doesn't matter who's in the uniform, it's the standard of play," said Battier, who had nine points and played a lead role in defending Joe Johnson, who made only 7 of 20 shots from the field.
"I've been in a lot of games in my 10 years in the league and this is right up there," Battier said.
The Hawks, whose victory at Miami gave the Heat their only loss, struggled with poor shooting.
There were 17 lead changes in the first half. Miami led 54-49 at halftime.
- The Hawks' last triple-overtime game was a 123-121 loss at Houston on Feb. 22, 2004.
- G-F Tracy McGrady played 29 minutes off Atlanta's bench after coach Larry Drew said the veteran's status was uncertain due to a bruised knee.
- The Hawks will play at Charlotte on Friday and play host to Chicago on Saturday to complete three games in three days.
- Heat F Mike Miller, recovering from hernia surgery, has been cleared for full contact. Spoelstra said he would ease Miller into action.