With the Rockets seemingly ready to rout the home team, the Warriors went to a smaller lineup and closed the first half on a 21-4 run. Shaun Livingston scored 14 of his 18 points in the quarter, helping Golden State go ahead 58-55 at halftime.
The Warriors held off Houston in the final minute of the fourth.
A picture truly is worth a thousand words pic.twitter.com/6tVYoWG85I— Golden St. Warriors (@warriors) May 20, 2015
James Harden, the runner-up to Curry in the MVP voting, nearly brought the Rockets back after Dwight Howard departed with a left knee injury. Harden finished with 28 points, 11 rebounds, nine assists and four steals.
Game 2 is Thursday night in Oakland.
Harden, serenaded with chants of "Over-rated!" from Warriors fans, mixed in a series of step-back jumpers and driving layups to help Houston even the score at 95-all midway through the fourth.
But the Warriors shut down Houston for long stretches, and Curry kept hitting shots. He connected on a 3-pointer and converted a layup to put Golden State up 108-97 with 2:01 remaining.
The Rockets never relented, though, with Trevor Ariza making a 3-pointer that trimmed the Warriors' lead to 108-106 with 14.6 seconds to play.
PHOTOS: Golden State Warriors fans 2015
Curry twice caught the inbounds pass, and the Rockets were forced to foul him both times. He hit both free throws to seal Golden State's win.
Curry added six rebounds and five assists, and Green had 13 points, 12 rebounds and eight assists to boost the Warriors when they needed it most.
Ariza scored 20 points and Josh Smith had 17 points and seven rebounds for Houston, which played without Howard for most of the fourth. The center's status moving forward is unclear.
In the conference finals for the first time since 1976, the Warriors hardly looked like the league's top-seeded team at the outset. Instead, the Rockets rode the momentum from a stunning 3-1 series comeback against the Los Angeles Clippers that ended with a Game 7 win in Houston on Sunday.
The only setback to Houston's hot start came when Howard briefly left in the first quarter after colliding with Smith. Howard returned after a few minutes, and the Rockets raced out to a 49-33 lead midway through the second quarter that left the home crowd stunned and silent.
PHOTOS: Golden State Warriors NBA playoffs 2015
That didn't last long.
With Andrew Bogut in foul trouble and the Rockets rolling on both ends, the Warriors put Green - the runner-up for NBA Defensive Player of the Year - at center and spread the court with shooters.
Livingston scored 14 points in the quarter, and Curry capped the spurt with a step-back 20-footer that sent the crowd roaring even louder. Fans later broke out in chants of "M-V-P!" as the Warriors sprinted to the locker room with a 58-55 lead.
They never trailed again.
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