All six of those overtime points came at the foul line, where Durant was 19-of-21 for the game. The Thunder went 34-of-39 from the charity stripe overall while ending the Clippers' six-game winning streak.
Los Angeles scored the final points of regulation on a Matt Barnes layup with 39 seconds remaining.
The Clippers became the last team in the NBA this season to lose a road game, having won their first three away from home.
Russell Westbrook added 23 points with nine assists and Serge Ibaka contributed 15 points for the Thunder.
Los Angeles was going after its first seven-game winning streak in 21 years.