OKC had applied the clamps through the first three quarters, holding San Antonio to 38.8-percent shooting (26 for 67) from the field and 16 turnovers. Thunder outscored the Spurs 24-16 in the third quarter to take its largest lead of the night.
"It was perseverance more than anything," said Spurs center Tim Duncan, who 16 points and 11 rebounds. "They were playing well in that third quarter and we stuck with it."
Manu Ginobili scored 11 of his team-high 26 points in the final period, going 3 for 3 from the field and 5 for 5 from the line.
He finished 9 for 14 from the field and 3 for 5 from 3-point range in Game 1. This came after shooting 25.7 percent (9 for 35) from 3-point range in the first two rounds.
"It just happened," Ginobili said afterward. "I don't know how exactly because I haven't scored like this all season long, but it happened and I'm very happy about it."
Led by Ginobili, San Antonio's bench outscored the Thunder bench 52-37 as guard Gary Neal pitched in 12 points and center Tiago Splitter had nine points and six rebounds in 12½ minutes.
"Oklahoma, they're a very good team," said Parker, who finished with 18 points, eight rebounds, six assists and two steals. "They made it very, very hard on us tonight."
Game 2 in Tuesday at 8 p.m. at the AT&T Center.
"I know if we play the same way, we're not going to win Game 2," Parker said. "We're going to have to play a lot better."