Dwyane Wade scored 28 with nine rebounds and Chris Bosh had 20 points and 12 rebounds for Miami, which could eliminate the defending conference champions and advance to the East finals with a victory in Game 5 in Miami on Wednesday night.
James, Wade and Bosh scored 83 of Miami's 98 points and had 35 of its 45 rebounds.
Paul Pierce scored 27, Ray Allen had 17 and Kevin Garnett had seven points and 10 rebounds for Boston.
James made up for a turnover at the end of regulation with a fallaway jumper to start overtime and give Miami the lead for good. The Celtics scored just four points in overtime, shooting 1 for 6 and making four of their 18 turnovers.
Rajon Rondo, who dislocated his left elbow Saturday night in Game 3, played 39 minutes with a padded sleeve covering what appeared to be a brace on his left arm, scoring 10 with five assists.
Boston took an 84-81 lead with 2:28 left after back-to-back 3-pointers by Delonte West and Allen, but James hit a 3 to tie it and then made a left-handed lay-in with 48 seconds to play to give Miami an 86-84 lead. Pierce drove to the basket to tie it, but James lost control of the ball while he was dribbling down the clock.
Allen went for it and was bumped by James Jones, giving Boston the ball out of bounds with 19.5 seconds left and the game tied at 86. Pierce got the ball, waited out the clock and then took a high-arcing fallaway jumper in the final seconds that bounced off the rim as time expired.
In the overtime period, Garnett threw the ball away and then James made a fallaway jumper as the shot clock expired to give Miami an 88-86 lead. Allen missed, Bosh dunked at the other end, West missed a 3-pointer and then James drew a charging foul on Pierce. After another Boston turnover, Wade made a 2-pointer with his foot on the 3-point arc to give Miami a 92-86 lead with two minutes left in overtime.
One game after the Heat's Big Three combined for just 44 points — their lowest total since they assembled their superteam in Miami — they provided almost all of the offense. No one else had more than four points, and the Heat's bench totaled just seven.
Miami won the first two games at home, but the Celtics responded when the series returned to Boston and clamped down on the Heat's Big Three to win Game 3, 97-81. Garnett had his best game of the series, scoring 28 with 18 rebounds, but the star for Boston was Rondo.
The Celtics point guard returned from a dislocated left elbow and provided his team with the spark it had been missing.
There was some doubt about whether he would be ready for Game 4 — but not in Rondo's mind.
An MRI and CT scan on the day off were both negative, and Rondo was back in the lineup.
Instead, it was Heat coach Erik Spoelstra shuffling his lineup, activating Udonis Haslem for the first time since November foot surgery and rewarding Joel Anthony for his play in the early games with his first start of the playoffs.
Haslem made just one appearance in the first three quarters, playing 3 minutes and picking up an offensive foul, a loose-ball foul and a technical foul.
But the Big Three was back.
The Heat went on a 10-2 run in the third quarter to take a 63-59 lead. It was a three-point game when Allen hit a 3-pointer and Boston scored 11 straight points to take a 73-65 lead before Miami scored the last two baskets — on layups by Anthony and James — to end the third quarter trailing by four.
Notes
Miami had zero assists in the second quarter, and just eight through three. … Boston activated point guard Carlos Arroyo in case Rondo's elbow and West's shoulder became a problem. He did not get into the game. … Jermaine O'Neal was called for a flagrant foul on James in the first five minutes, perhaps an attempt by the officials to make sure things didn't get nasty after Saturday night's takedown in which Rondo was injured.