The defending champion Heat took a 2-1 lead in an Eastern Conference semifinal series marked by technical fouls and ejections.
James was involved in a heated exchange with Nazr Mohammed in the second quarter, one that led to Mohammed's expulsion from the game, but the Miami superstar shook off that incident to come through down the stretch.
Chicago trailed by only four when James hit a 3-pointer with 2:35 remaining and he followed that with a layup and 10 free throws the rest of the way.
James made just 6-of-17 from the field, but hit all 11 of his free throws while grabbing eight rebounds and handing out seven assists.
Chris Bosh added 20 points and 19 rebounds for the Heat and Norris Cole scored 18 points off the bench.
Chicago was led by Carlos Boozer with 21.
The second-period confrontation came when Mohammed put an arm around James to impede his progress in the frontcourt. James wrestled free from Mohammed's grasp and was immediately called for a foul.
James threw up his hands in protest that the foul had been called on him and as he did so he was shoved to the floor by Mohammed. That earned Mohammed a trip to the locker room for the duration of the game and he could face further punishment from the NBA.
Chicago, which surprised Miami in the opening game of the series on the Heat's home floor, will host Game 4 on Monday.