Draymond Green gave Golden State the lead for good on a three-point play after the Warriors trailed by as many as 25 points.
Leonard left in the third quarter after he re-injured his left ankle, and the Warriors capitalized with an 18-0 run.
Kevin Durant scored 10 straight points during a key fourth-quarter stretch on the way to 34 points, while
Leonard had 26 points and eight rebounds.
Somehow, these unusually careless, cold-shooting Warriors found a way to keep their unblemished postseason record alive at 9-0.
Game 2 in the series is Tuesday night back at Oracle Arena.
Curry scored 19 points in the third quarter and made 14 of 26 from the field overall, including seven of 16 long range.
"It's the playoffs. You've got to expect everything," Curry said. "I wouldn't call it smooth sailing at all. We've actually had to execute and get to this point. They came out and challenged us heavy, put a nice little run together.
"It took us a minute to figure it out to get up to that game speed after this little break. It's definitely a nice way to win Game 1. We've got to capitalize off that and start Game 2 a lot better."
Leonard first went down after stepping on a teammate's foot along the bench after taking a shot. Pachulia later slid into him as he came down from shooting and Leonard fell into the Spurs bench, grabbing his ankle.
Losing Leonard could be a story of this series. He made seven of 13 shots in 23 minutes to go with eight rebounds and three assists. He converted all 11 of his free throws.
He sat out Game 6 of the last series against Houston, a 39-point, series-clinching win on Thursday, after injuring the ankle in a Game 5 overtime victory.
Durant has repeatedly spoken of elevating his game when it matters most — and wow did he do it at the very moment the Warriors needed a huge lift on their home floor.
His three-pointer with 7:53 to play got Golden State within 94-88, then he drove to his left and right by Kyle Anderson for an emphatic one-handed slam the next possession to ignite a deafening sellout crowd. Durant made another long-range shot at the 6:10 mark and a jumper the next time down before
"Steph got us going in the third, I just tried to do my part in the fourth," Durant said.
Gregg Popovich lost to good friend and old pupil
Yet for much of the afternoon, the Spurs outplayed the mighty Warriors on both ends of the floor.
While Kerr spoke to the team at halftime, he watched the game from behind the scenes in the arena a day after returning to the practice floor for the first time in more than three weeks.