Manu Ginobili scored 11 of his 26 points in the fourth quarter and host San Antonio stormed back from a nine-point deficit to win the opener of the Western Conference finals.
Tim Duncan had 16 points and 11 rebounds as the Spurs improved to 9-0 in the postseason and won their 19th consecutive game.
Tony Parker scored 18 points and Gary Neal had seven points during an 18-3 fourth-quarter burst as San Antonio held Oklahoma City to one field goal in a span of over nine minutes.
Game 2 is Tuesday night in San Antonio.
Kevin Durant scored 27 points to pace the Thunder, who led 73-64 with just over 11 minutes to play but were outscored 39-27 in the final quarter. James Harden added 19 points and Russell Westbrook had 17 but each had miserable shooting nights, combining to go 14 for 38 from the floor.
The Thunder appeared primed to steal Game 1, closing the third quarter on a 10-2 run to open its biggest lead at 71-62 entering the final 12 minutes.
The Spurs charged back and went ahead 82-76 on Neal's 3-pointer with 5:05 to play. The Thunder were still within 88-84 with 3:17 to play, but Stephen Jackson drilled a 3-pointer and Ginobili converted a three-point play to build a 10-point lead.
After shooting 25 percent (6 of 24) in the third quarter, San Antonio hit nine of its first 11 shots in the fourth to ignite its comeback.
Ginobili scored the final seven points of a 10-2 run, including a 3-pointer at the buzzer, to give the Spurs a 24-18 lead after a sloppy opening quarter. Derek Fisher came off the bench to spark the Thunder in the second quarter, scoring nine points on 4-of-4 shooting to help Oklahoma City take a 47-46 lead into the locker room.
GAME NOTEBOOK: The Spurs matched the Los Angeles Lakers' record 19-game winning streak carried from the regular season into the playoffs. The Lakers set the mark in 2001. The Spurs have not lost since April 11. ... Durant averaged 22.7 points in three regular-season meetings against the Spurs, more than five points below his league-leading 28-point average. ... San Antonio has won nine of the past 10 meetings with the Thunder. ... Westbrook was attended to by the team trainer after he crashed to the floor with 2:46 to play on an offensive foul. He stayed in the game.