Green hit seven of nine 3s, Neal was six for 10 from deep and Tim Duncan chipped in 12 points and 14 rebounds for the Spurs, who responded to a 19-point drubbing in Game 2 with a blowout of their own.
The Spurs flummoxed four-time MVP LeBron James for a third straight game, holding him to 15 points on 7-of-21 shooting. James also had 11 rebounds and five assists, but he missed 11 of his first 14 shots and never looked comfortable against San Antonio's swarming defense.
Game 4 in the best-of-seven series is on Thursday night in San Antonio.
Kawhi Leonard had 14 points and 12 rebounds for the Spurs, who are two wins away from the franchise's fifth championship with two more games at home in the raucous AT&T Center this week.
Mike Miller made all five of his 3-pointers for 15 points and Chris Bosh had 12 points and 10 rebounds for the defending champion Heat, who never led in the game.
The Spurs came home to their devoted fans encouraged by the split, but reeling a little after James, Dwyane Wade and Mario Chalmers thumped them 103-84 in Game 2.
When the series started, it was billed as the battle of the Big 3s -- James, Wade and Bosh for Miami against Tony Parker, Duncan and Manu Ginobili for the Spurs. But it's really been about the little guys for the most part, outside of Parker's sensational performance in Game 1.
Green, who was cut twice by the Spurs and also played in Slovenia, Austin and Reno in a roundabout journey to the NBA, was the lone bright spot in Game 2, hitting all six of his shots for 17 points.
Chalmers scored 19 points in Game 2 and ignited the game-deciding 30-5 run that blew the Spurs out of South Beach and swung the momentum back in favor of Miami, and Miller has been hitting everything he puts up.
Neal, a backup guard from Towson University, became the latest no-name to make an impact when he three 3s in the second quarter to put San Antonio in control. His 3-pointer just before the halftime buzzer gave the Spurs a 50-44 lead at the break, allowing the Spurs to weather a quiet start from Parker, who had just six points. Parker briefly left the game going back to the locker room in the third quarter before returning at the start of the fourth quarter.
It was Miller's shooting, not James' overall brilliance or Wade's tenacity, that kept it from turning into a blowout in San Antonio's favor earlier in the third quarter. He hit two from long distance midway through the period to keep Miami within 10, but Neal added a runner and Ginobili threw down his second dunk of the game to push it back to 14 in a heartbeat.
James has had an unusually difficult time scoring in these finals, entering the game averaging a pedestrian 17.5 points on 41 percent shooting. He posted his two lowest scoring outputs -- 18 and 17 -- in the two games in Miami, with the Spurs forcing him to be a jumpshooter and determined not to let him attack the rim.
James has been content getting others involved, preferring to find Wade, Miller and his supporting cast for open shots rather than force anything like he used to do in Cleveland. He and Bosh started out a combined 3 for 11, and the four-time MVP appeared to grow frustrated late in the second quarter. He slammed both hands on the press table after a 1 for 6 start, then got a layup in transition with just over two minutes to play in the half for his first points of the second quarter.
He showed a flash of getting going with six straight points late in the third quarter, but Neal hit two 3s out of the fourth quarter gates, Green hit a couple himself and the lead swelled to 94-65 to end all doubt.
It was the first time since 2007 that a finals game was played in San Antonio, a proud military city that had grown accustomed to hosting the NBA's biggest event in the middle of the last decade. The Spurs won three titles in five seasons during that span, and the core of Parker, Duncan and Ginobili were back for one more this time around.
They've waited a long time to see another one -- Parker said it felt like "forever" to him -- and let their voices be heard early in Game 3.
The crowd chanted "Go Spurs Go!" as the lights dimmed before the game, a familiar refrain that could be seen all across town this week -- on the backs of taxi cabs, hanging from banners on buildings and even being chanted by kids appearing on a local morning show.
The uncommon stability in the organization has struck a cord with the fans here, who have watched their core grow up and grow old right in front of their eyes. They watched from afar as their team squeaked out Game 1 in Miami thanks in large part to an incredible shot by Parker in the closing moments.
James and the Heat responded emphatically in Game 2, ripping off a 30-5 run that spanned the third and fourth quarters to blow the Spurs out and snatch all the momentum.
If any team was prepared to enter the hostile AT&T Center and get a victory, it was the Heat. They've won on the road in every series of the playoffs and have been the target of intense scrutiny ever since James, Wade and Bosh united three years ago to chase championships.
They already have one championship in hand and a victory in Game 3 would go a long way toward a second. Of the last 12 NBA Finals to split the first two games, 11 of the Game 3 winners have gone on to capture the trophy.
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