The Heat took a 3-1 lead in the best-of-seven series two days after having their eight-game postseason winning streak snapped on Saturday in a 104-90 loss.
James missed a free throw with 1.1 seconds remaining and a chance to score 50 points for the first time in the playoffs.
He scored at will, but teammate Chris Bosh hit the biggest shot -- a 3-pointer from the right corner with 57.3 seconds remaining that gave Miami the lead for good.
Game 5 is Wednesday in Miami.
Notable deaths of 2014 [PHOTOS]