Chris Bosh finished with 18 points and Rashard Lewis and Dwyane Wade each scored 14 points for the two-time defending NBA champion Heat, who knotted the championship series with the top seeded Spurs at one game apiece.
“I was happy I was able to make some plays to help us get the victory tonight,” said James. “I just continued to attack.”
The best-of-seven series shifts to Miami for game three on Tuesday.
James, who had his toughness questioned after game one, helped the Heat maintain their streak of following up playoff defeats with victories.
They have now rebounded from post-season defeats with victories 13 straight times, the longest active streak of its kind in the league.
“We have been through enough of these situations that you have to play through all the ups and downs and you have to play through the doubt,” Miami coach Erik Spoelstra said. “You have to keep an even keel while somehow keeping a ferocity to your play.”
James connected on 14-of-22 from the field in 37 minutes of playing time, and was one of the biggest factors when the game was on the line in the fourth quarter.
It is a role James has played consistently throughout his career but one he was prevented from performing in a 110-95 loss in game one after the air conditioning system at the AT&T Center failed, contributing to his dehydration and cramps.
The climate control was back to normal on Sunday, with temperatures in the arena hovering around 21 degrees Celsius (70 Fahrenheit), compared to 30-plus on Thursday.
“He has an incredible way to put his fingerprints on a game,” Spoelstra said of James.
Tim Duncan paced the Spurs attack with 18 points and 15 rebounds. His 157th career playoff double-double tied him for the most all-time with Magic Johnson but wasn’t enough to hold off the Heat.
Duncan has scored in double figures in 222 of his 231 career playoff games.
Tony Parker finished with 21 points and seven assists while Manu Ginobili came off the bench to score 19 points for the Spurs, who are trying to win their fifth NBA title since 1999 after losing in seven games to the Heat last season.
San Antonio committed 11 turnovers on Sunday for a total of 33 in the first two games of the series.
– ‘Ball stuck to us’ –
“LeBron did a great job at his end and we had to be really perfect at the other end and we didn’t,” said Spurs coach Gregg Popovich. “The ball stuck to us. We tried to do it individually and we are not good enough to do that.”
This is the first NBA finals rematch since the Chicago Bulls battled the Utah Jazz in 1997 and 1998.
The Heat lost the finals in 2011 then won back-to-back titles in 2012 and last season.
Bosh rattled in the go-ahead basket with 78 seconds to go, connecting on a 23-foot jumper from beyond the arc to give the Heat a 95-93 lead.
Bosh then made a nice feed to Wade in close for a field goal to give Miami a five point cushion with nine seconds left.
Ginobili drained a three pointer at the buzzer to cap the scoring as the time ran down on the Spurs.
San Antonio had an 11-point lead in the second quarter but then James went to work. He scored 11 of his 13 first-half points in a second quarter in which Miami outscored the Spurs 24-17.
The game was tied 43-43 at the half and the Spurs led 78-77 at the end of the third in a back-and-forth contest that included 17 lead changes.
Spoelstra said his players have to guard against getting overconfident after stealing a victory on the road before heading home for the next two.
“We had two days to commiserate how game one went down,” he said. “It was frustrating and painful going through that for two days and now we have to manage the other emotion. That can be just as challenging.”