Nate Robinson had 27 points and nine assists and hit the go-ahead shot with 1:18 remaining while his Chicago team-mates Jimmy Butler and Joakim Noah had 21 points and 14 rebounds and 13 points and 11 boards respectively.
The fifth-seeded Bulls had a dominating 46-32 rebounding edge and again played without forward Luol Deng (illness) and point guard Kirk Hinrich (calf).
LeBron James scored 15 of his 24 points in the final quarter and Dwyane Wade added 14 points for the Heat, who shot 39.7 percent from the field. Game 2 is Wednesday in Miami.
The Heat had a 76-69 with 6:30 remaining after back-to-back three-point plays by James. Miami's lead was three before Marco Belinelli hit a tying three-pointer with 1:59 left to start a game-ending 10-0 run that featured Robinson scoring the game's final seven points.
In the night's other game, Manu Ginobili buried a three-pointer from the left wing with 1.2 seconds left in the second overtime as San Antonio came from behind to stun visiting Golden State 129-127 (2OT) in Game One of the Western Conference semi-finals.
The Spurs trailed by as many as 18 points at one stage and were 16 behind with less than four minutes left in the fourth quarter before a furious rally forced overtime.
Tony Parker led the way with 28 points, eight assists and eight rebounds and Danny Green knocked down six three-pointers en route to 22 points as San Antonio won its fifth straight game to begin the playoffs.
Stephen Curry played all 58 minutes and collected 44 points and 11 assists, including the pass that led to Kent Bazemore's go-ahead layup with 3.9 seconds left in the second overtime that gave the Warriors a brief lead.
Harrison Barnes added 19 points and 12 rebounds and Klay Thompson scored 19 before fouling out during the Spurs' rally at the end of the fourth.
The Warriors led 104-88 late in the fourth when Parker scored six straight points and scooped in a transition layup to highlight a 15-0 run that cut the deficit to a single point with 59 seconds to play.