The experienced Spurs, who have won four NBA championships under coach Gregg Popovich, trailed by nine after three quarters but kept their cool and sprinted to the finish to win the opening game of the best-of-seven series.
Manu Ginobili scored a team-high 26 points off the bench, and Tony Parker added 18 as San Antonio outscored the visiting Thunder 39-27 in the fourth to pass a rare post-season test.
“"They had us on our heels and we were not aggressive," Ginobili told reporters of the game's first three quarters. “We needed an edge. In the (fourth quarter), we had it."
The red-hot Spurs remain perfect in the post-season with nine victories and have now tied the 2001 Los Angeles Lakers for longest win streak between the regular season and playoffs.
Kevin Durant netted 27 points and 10 rebounds for Oklahoma City, who defeated the past two NBA champions in their previous rounds and were in good shape to move ahead of San Antonio before getting blitzed in the fourth.
Three-time defending scoring champion Durant managed no field goals in the last 16-plus minutes of the game.
“"They forced me out a little bit and they do a good job on help defense," Durant said. “"I did things I wanted to do tonight, I just have to do better next time."
San Antonio opened the final period with a 15-3 run to seize a lead they would never relinquish.
The home team built a 10-point advantage with two minutes remaining, and James Harden trimmed the final scored with a trio of three-pointers in the last couple minutes.
Harden finished with 19, and Russell Westbrook had 17 but only made 7 of 21 shots.
Duncan chipped in 16 and 11 rebounds for the Spurs, who also got a boost from reserve Stephen Jackson's late defense against Durant.
“"It's a long series and you have to continue to play," said Oklahoma City guard Derek Fisher, who had 13 points. “
"They showed mental toughness tonight in being able to battle back, and now it's our turn."