Although LeBron James scored 30 points, the Heat fell behind in the first quarter and were never able to recover as Chicago improved to 18-13 in the Eastern Conference.
Miami still lead the East with a 22-9 record but have stuttered recently, having lost three of their last four games.
James went into the game with doubts over a knee injury but still played more than 41 minutes and maintained his record of scoring at least 20 points in each game this season.
Four times previously in this campaign, Miami have been out-rebounded by 15 or more but on each occasion found a way to win.
However, they were unable to repeat the feat against the burly Boozer and a determined Bulls, who demolished the Heat 19-4 in offensive rebounds.
"They pounded us on those second chance points. They were just throwing it up there and playing volleyball against us," Heat head coach Erik Spoelstra said.
"Everybody understands in both locker room what the overwhelming key to the game was and we've got to get better," he added.
"It's not the first time we absolutely got pounded by Chicago on the glass. They do it to a lot of teams and they've done it to us before.
"You have to come in with a 'gridiron' mentality against them. They had us on our heels the entire night on that backboard."
The Bulls are still without long-term absentee Derrick Rose, who has been sidelined since suffering a serious knee injury in the opening game of last season's playoffs.
Despite missing their key player, the Bulls are 8-1 on the road this season against Eastern Conference opponents and restricted the Heat to their lowest points this season.
"They are the defending champions and a great team. In order to beat them, you really have to beat them, they don't give you anything," Bulls coach Tom Thibodeau said.
"I thought we fought hard and battled. We understand how important rebounding is and it showed tonight. I thought that was the difference in the game."
The Heat know they will have to find a solution to their rebounding weakness before the playoffs but an answer to the issue appears to remain elusive.
"I'm out of recommendations," Miami center Chris Bosh said.
"We just have to keep fighting, it is not the first time we have been out-rebounded by 20. Who says that and is in first place in the East? I'm just trying to figure it out myself."
The nearest Miami came to a rally in the fourth was when James was fouled after a steal, with what was ruled clear-path foul by Chicago's Kirk Hinrich with 90 seconds to play.
James went to the line, with the Heat allowed to keep possession and he made one of the two free throws to bring the Heat to within five points but a turnover gave the ball back to the Bulls and assured them of victory.
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