The Thunder were down by 10 as the fourth quarter began and San Antonio quickly increased their lead to 12, before the home side began chipping away at the deficit.
Russell Westbrook capped that with a couple of clutch free throws in the final 10 seconds, leaving Manu Ginobli to miss a potentially series-winning buzzer beater in regulation.
That miss didn't prove costly for the Spurs as the players' experience and the direction of legendary coach Gregg Popovich proved too much for a misfiring Thunder to handle.
Adams didn't feature in the fourth and he was also inactive during overtime, being left on the bench after some uneven play in the first three quarters. The centre finished with two rebounds in 12 minutes of court time, missing his only shot in the first and turning over the ball in the third.
His limited contributions will hardly detract from his remarkable rookie year, though, with a player some thought destined for the NBA's development league eventually becoming an integral member of a title challenger.
After playing some of his best basketball in the postseason, Adams looks set for a bright future on a team that will contend again next season. Depending on the fate of centre Kendrick Perkins, frequently a target for criticism among Thunder fans, Adams could even find himself in the starting five.
The question now will be whether he will represent the Tall Blacks at September's Basketball World Cup, with the Rotorua product set to make a decision in the offseason.
For New Zealanders there is some consolation in the Thunder's exit, with Sean Marks, still the only Kiwi with an NBA title, among the Spurs' coaching staff.
