The Thunder eliminated the Spurs in six games, capturing the Western Conference title and setting up a fight for the championship against either Boston or Miami.
Boston will take a 3-2 advantage into its meeting with the Heat Thursday.
San Antonio appeared ready to force a seventh game in the Western Conference title series, jumping to as much as an 18-point lead during the second period and still owning a 17-point advantage at intermission.
Oklahoma City overcame that deficit by late in the third quarter, however, and pulled away down the stretch to bring the Spurs' season to a frustrating close.
San Antonio won its final 10 games of the regular season and then captured its first 10 playoff contests -- including the first two games of the conference finals -- before losing four in a row to the Thunder.
Durant made 9-of-17 from the field and 12-of-15 from the line. He produced 34 points, grabbed 14 rebounds and handed out five assists.
Russell Westbrook added 25 points along with eight rebounds and James Harden had 16 points off the bench. Oklahoma City shot 50 percent from the field and made 10-of-18 from 3-point range -- four of the long-range shots coming from Durant.
Parker had 29 points and 12 assists, Duncan scored 25 points with 14 rebounds and Jackson had 23 points.
San Antonio still had a chance to pull out the game heading into the final minute, trailing by four and having possession.
A Duncan layup attempt, however, was blocked by Kendrick Perkins and even though the Spurs came away with the offensive rebound they suffered missed 3-point attempts by Jackson and Kawhi Leonard.
Harden finally claimed an Oklahoma City rebound, leading to a dunk by Kendrick Perkins with 25 seconds to go that all but clinched the outcome.