The best-of-seven series moves to Chicago for Game Five on Thursday and that promises to be another bruising but entertaining encounter in what has been a thrilling series.
A compelling and physical game finished 85-85 in regulation after league MVP Derrick Rose missed two late opportunities including a game-winning shot for the Bulls with the final play.
Rose had the game in his hand after an offensive foul was called on Miami's LeBron James with eight seconds left - but he was unable to find the basket.
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"Tonight was definitely on me. I had two opportunities to end the game. I couldn't do really was my fault but I am going to learn from it," said Rose, who made 23 points but gave up seven turnovers.
"It is not over yet - we still have games to play, we are going home, we have to be positive and they are beatable."
The MVP's game was made more difficult by James opting to mark him in the crucial latter stages and overtime, with the attention making Rose's final shot in regulation so difficult.
Bulls coach Tom Thibodeau was quick to give his backing to the 22-year-old Rose.
"He missed but I wouldn't want to have any other player, he is a great player and a great competitor and I have great belief in him" he said.
The Heat, with James (35 points) taking charge on both sides of the ball, dominated in overtime and extended their post-season home record to 8-0.
The loss was the first time this season the Bulls have lost three successive games.
Dwyane Wade had a quieter than normal game for Miami until overtime when he suddenly found the touch and rhythm that had been absent.
He produced a crucial jump shot to make it 91-88 and then made a driving lay-up on a turnover to extend the lead to six points before making a key block.
"The most important thing is that he was able to make game-winning plays at the end. It took great mental fortitude for Dwyane to leave the rest of his frustrations behind and find a way to help us win," said Heat coach Erik Spoelstra.
While the focus was inevitably on the final stages of the game, the encounter had ebbed and flowed from the outset with both teams enjoying spells of dominance where they showed their strengths.
The Bulls led by 11 points in the first quarter but then James led a 14-0 run from Heat that got them back into game.
Chicago responded though with Rose making two sensational dunks as the Bulls fought their way to lead 46-44 at halftime, which they continued in the third to take 68-63 lead before the dramatic fourth unfolded.
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