Green scored 11 of his 13 points in the final period as the Spurs hit 14-of-16 shots, including all six from beyond the arc, turning a seven-point deficit into a runaway victory.
In the final quarter, Green made all three of three-point attempts to key a 31-9 run to end of the game.
Tim Duncan, who led the Spurs with 21 points, said taking care of the ball made the difference down the stretch. San Antonio had 22 turnovers but was dazzling when it counted.
"We took care of the ball finally," he told reporters. "We had 20-some odd turnovers for 28 points. The last seven or eight minutes we only turned the ball over once or twice.
"Getting them out of the passing lanes, keeping them in front of us, making them score over us, and we found something that worked on the offensive end.
"We started moving the ball and Danny Green got hot and hit a couple of shots and really opened it up for us."
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.
"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."
Tony Parker scored 19 points and Manu Ginobili added 16 for the Spurs, who host Miami in Game Two of the best-of-seven series on Sunday.
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 there."