The Thunder roared back from an 18-point deficit in the first half, outscoring San Antonio 59-36 after the intermission to claim the best-of-seven Western Conference series 4-2 on their home court.
Oklahoma City will now play the winner of the Eastern Conference final between the Boston Celtics and Miami Heat in the best-of-seven NBA championship series.
After halting the Spurs' 20-game winning streak last week, Oklahoma City surged into the championship series by taking four straight games to overturn a 2-0 deficit.
Three-time NBA scoring champion Durant, 23, led the charge with a game-high 34 points and 14 rebounds while playing the entire 48 minutes.
"This is the toughest game I've played since I've been here," Durant said in an on-court ceremony as the packed home arena crowd interrupted him with a chant of "MVP, MVP, MVP."
"But we kept playing hard. I tried to inspire my team mates with my play on both ends of the floor, and I'm glad we got this victory for Oklahoma City."
Durant and 23-year-old point guard Russell Westbrook helped bring the Thunder to the NBA Finals in the franchise's fourth season in Oklahoma City after the move from Seattle.
The Spurs, a veteran team with core players Tim Duncan (36), Manu Ginobli (34) and Tony Parker (30), came out on fire, determined to force a Game Seven back in San Antonio.
French point guard Parker lit up the Thunder with 21 points and handed out 10 assists in the first half, with Duncan adding 13 points as the Spurs held a 15-point lead at halftime 63-48.
Durant said the half-time message to the Thunder was clear.
"The coaches said we got to play with force, with a sense of urgency and we didn't want to go back to San Antonio," he said.
"Leave everything out on the floor."
The Thunder went on a 14-4 run to close to 67-62 and grabbed a 79-77 lead on one of Durant's long three-point bombs from well beyond the arc late in the quarter.
Parker, silenced for most of the third period, came to life at the end of the stanza, scoring on a spinning drive to the hoop to give San Antonio an 81-80 edge going to the last quarter.
Durant and Westbrook, however, helped the Thunder build a six-point lead at 90-84 and San Antonio could not catch the younger, faster team.
"They were more energetic," said Spurs guard Manu Ginobli.
"They played better defence, they were more aggressive.
"Of course, we can't have their legs, their energy. We're never going to jump as high or run as fast."
Veteran guard Derek Fisher, added to Oklahoma City's young roster during the season, hit a critical three-pointer late in the game and banked in a jumper to help keep the Thunder ahead.
"We're just playing and doing it for each other, and great things can happen when you do it for each other," Fisher said.
Westbrook, 23, added 25 points for Oklahoma City.
Parker finished with 29 points and 12 assists, but just eight points and two assists coming after intermission.
Tim Duncan contributed 25 points and 14 rebounds, while Stephen Jackson came off the bench to score 23 points, going 6-for-7 from three-point range for San Antonio.
The Celtics host the Heat on Thursday, with Boston leading the series 3-2. Game One of the NBA Finals will be staged in Oklahoma City on Tuesday.
As elated as the Oklahoma City players were, coach Scott Brooks reminded his team there was more work ahead.
"We don't want to just get to the finals," said Brooks. "We want to win championships."