The Spurs lost star forward Kawhi Leonard to a recurrence of his left ankle injury in the third quarter after having built a 25-point lead in the first half.
Game two in the best-of-seven series is Wednesday, also on the Warriors' home court.
Rallying from a 20-point halftime deficit, the third-biggest comeback in NBA history, the Warriors got a game-high 40 points from Stephen Curry and 34 from Kevin Durant.
Curry hit a 13-foot floater with nine seconds remaining to give Golden State a 113-110 lead and begin a hectic finish.
LaMarcus Aldridge, the Spurs' leading scorer with 28 points, had a chance to tie the score but misfired on a three-pointer from the left corner.
Patty Mills retrieved the ball and was fouled with five-tenths of a second left. He made the first free throw to get the Spurs within two, then intentionally missed the second, with the Warriors deflecting the ball out of bounds as time expired.
Curry scored 21 of his points on 7-of-16 shooting on three-pointers. The Warriors were 11 of 30 from beyond the arc and outscored the Spurs 33-21 on three-pointers.
Leonard finished with 26 points and Manu Ginobili 17 for the Spurs.
The game turned on Leonard's injury. He had poured in 18 points and Aldridge 17 in a stunning first-half offensive display by the Spurs, who led by 25 at one point en route to a 62-42 halftime advantage.
Leonard, who missed the end of Game Five of the Western Conference semifinals and all of Game Six against the Houston Rockets, tweaked the ankle when he stepped back into the Spurs' bench while attempting a three-pointer early in the third quarter.
Then after a brief stint on the bench, he returned and landed on Warriors center Zaza Pachulia's foot on a jumper, which sent Leonard limping to the bench for good.
The Spurs were leading 78-55 when Leonard departed with 7:52 remaining in the third quarter. The Warriors immediately responded with an 18-0 run to get back into the game.
