San Antonio trailed by seven in the final period but out-scored the Miami 31-9 over the last nine-plus minutes. The Spurs drained six three-pointers, including three by Green, over the last six minutes.
LeBron James paced the Heat with 25 points but played only five minutes in the final quarter due to leg cramps.
An electrical outage caused the air conditioning to fail at the AT&T Center, creating steamy conditions for both players and fans. James hit a lay-up with four minutes left to pull Miami within 94-92 but was forced to the bench with his leg issues.
San Antonio were led in scoring by Tim Duncan's 21 points and Tony Parker's 19. LeBron's 25 was followed by Dwyane Wade with 19 and Chris Bosh with 18.
"It sucks at this point in time in the season," said James. "After I made that layup we was down two and, you know, as well as they played we still had a chance.
"After I came out of the game, they kinda took off. And it was frustrating sitting out and not be able to help our team."
Tim Duncan explained how the Spurs got the win: "We stopped turning the ball over. All of a sudden it turns into us with ball movement and our shooters getting open shots, which we were trying to get to all game long but they were too much in the passing lanes.
"Their defense was too aggressive and we just couldn't find a way, but we kept pushing and found a way."
Fifty-eight percent shooting, including 13-of-25 from beyond the arc, helped the Spurs overcome a 22-turnover performance.
The second game of the finals, a re-match of last year won by the Heat in seven games, will be in San Antonio on Sunday, before the series shifts to Miami for the next two games.
Miami coach Erik Spoelstra said the hot conditions created an "unusual environment," adding that it was painful to see James hobbled.
"We're used to having the hotter arena at this time of year," he said. "But both teams had to deal with it. It's unfortunate that it was that way.
"It felt like a punch in the gut when you see your leader limping like that back to the bench, but at the same time we still had an opportunity to make plays going down the stretch.
"They made obviously the biggest plays in the last five minutes."
With James on the bench, the Spurs out-scored Miami 16-3 over the last four minutes.
"Obviously when LeBron went out for us in a two-point game is not ideal, but we got to get better shots on the offensive end," said Miami guard Dwyane Wade, who had 19 points.
"We didn't, we turned it over a few times. We didn't get good looks and you can't do that versus this team."
Manu Ginobili, who had 16 points for San Antonio, said it was hardly time to be overconfident, noting that the Spurs won the opener in last year's finals before falling in seven games.
"At this point, we are the same exact spot as last year," he said. "We are up 1-0 and they always come back. They always adjust well.
"They have great players, so we got to come back on Sunday and play even better."
James said he welcomes the rest until Sunday's game.
"I need it, I need it, I need it," he said. "We're going to start tonight, continue to get the fluids in me and get me ready for Sunday.
"Thankful for the fact that I can get on it tonight, and put myself in a position where I can be out there for my team for the long haul.
"Sitting on the sideline, if I'm not in foul trouble, is not good for us and not good for me. I look forward to Game Two and go from ther
Spurs coach Gregg Popovich called the turnovers his team's "Achilles heel."
"When we have a game like that turnover-wise, invariably it's a loss for us," he said. "I haven't looked at any stats, I'm afraid to, but I guess we probably made threes.
"We made a decent number of threes. I think we rebounded pretty good from where I was sitting. But the turnovers are usually a killer. We feel very fortunate to have won this game tonight."
San Antonio led 54-49 at the half behind 15 points from Duncan, who hit six-of-seven from the floor to the delight of the sold-out crowd at the AT&T Center.
The Spurs held a 69-67 lead with just under four minutes left in the third quarter but veteran guard Ray Allen scored six points of an 8-0 Miami run to give the Heat a 75-69 lead.
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