The Celtics cut their deficit in the Eastern Conference finals to 2-1 with Game 4 set for Saturday in Boston.
Coming off an overtime setback in the second game of the series, Boston broke things open by holding the Heat to 14 points in the second period. The Celtics owned a 22-point lead heading into the final quarter and pulled off the critical win while shooting 50 percent from the field.
Garnett hit 10-of-16 shots and scored 28 points, Paul Pierce added 23 and Rajon Rondo had 21 points to go with 10 assists.
James took a third of Miami's shots while scoring 34 points. Dwyane Wade scored 18.
Miami owned a 28-22 lead with 2:42 left in the opening period, but did not score for the rest of the quarter or during the first 4:30 of the second. Boston ran off 15 straight points during the Miami drought to take control of the game.
The Heat continued to play without Chris Bosh, who missed his seventh straight playoff contest with an abdominal muscle strain. It has already been announced that Bosh will not play in the fourth game of this series.
Miami matched Boston in field goal totals, but made only 10-of-20 from the foul line and grabbed just six offensive rebounds.