James, who was named playoff Most Valuable Player for the second year in a row, also nailed five 3-pointers and had four assists for the Heat, who won a third title to go with crowns in 2012 and 2006.
"It feels great. This team is amazing," James said. "The vision I had when I decided to come here has all come true. To win back-to-back championships is an unbelievable feeling."
Dwyane Wade added 23 points and 10 rebounds and Shane Battier came off the bench to score 18 points on 6-of-8 shooting from beyond the arc for the Heat, the first team to win back-to-back titles since the Los Angeles Lakers in 2009 and 2010.
James earned his second championship ring for the Heat, who posted a league-best 66-16 regular-season record that included a 27-game winning streak, the second longest in league history.
It was a heartbreaking defeat for the Spurs and their first loss in five trips to the best-of-seven championship final.
San Antonio was just seconds from winning the NBA crown in game six but the Spurs blew a five-point lead late in regulation and then lost in overtime to set up the deciding game.
"Being so close and feeling that you are about to grab that trophy and then seeing it vanish is very hard," said the Spurs' Manu Ginobili. "I think if we would have lost both games like this (Thursday) I would have been a little more up."
James drained a key 19-foot jump shot with 28 seconds remaining after a Tim Duncan miss at the other end to give the Heat a 92-88 lead.
James then stole the ball from the Spurs' Ginobili and helped seal the victory with two free throws with 23 seconds to play, giving Miami a 94-88 lead.
Wade, who combined with James to score 60 of the Heat's 95 points, capped the scoring Thursday by making a free throw with 16 seconds remaining.
Heat coach Spoelstra said James' jump shot was the difference.
"It is unique for a guy who has been the best in the game since he was in the seventh grade; usually you wouldn't have that type of work ethic that would match that type of talent," Spoelstra said.
"As the series went on, he realized that was the shot that was going to be open and in the biggest game, the biggest moment, those are the shots that he hit. And those were the difference tonight."
Duncan had 24 points and 12 rebounds and Kawhi Leonard finished with 19 points and 16 rebounds for the Spurs, who were trying to become the first team since 1978 to win a seventh game of the NBA Finals on the road.
"It is no fun to lose," said Spurs coach Gregg Popovich. "But we lost to a better team.
"We just didn't play well. And a lot of that had to do with their pressure, their defense, a lot of contested shots. Their athleticism was just tough for us to handle and it finally showed a little bit in the seventh game."
Both teams shot poorly in the first half, especially the Spurs, who hit only 35 percent from the field. The Heat had eight turnovers and shot just 5-of-14 from beyond the arc in the first two quarters.
James and Wade combined to score 21 of the Heat's final 28 points in the second quarter.
Wade got hot from the opening tip and carried it into the second quarter. He scored 14 points on 7-of-12 shooting and had six rebounds as Miami took a 46-44 lead at half-time.
Duncan had 13 points while Ginobili scored seven in the first half for the Spurs.
James hit back-to-back 3-pointers late in the third quarter to give Miami one of their biggest leads to that point at 62-57. James finished a superb 5-of-10 from beyond the arc in game seven.
"He's the best conditioned athlete in this game," Spoelstra said of James. "He takes pride and puts so much time into it. He probably lost 12 to 15 pounds in this playoff run expending so much energy."