The Thunder went on a 14-4 run in the fourth quarter to pull within two points at 99-97 with just over 7 minutes to play.
The score was tied at 105 with about 2 minutes left when Serge Ibaka grabbed the rebound of a Russell Westbrook shot and launched a pass to Durant at the top of the key.
He then buried a 3-pointer with 1 1/2 minutes to go, putting Oklahoma City ahead to stay.
Durant also provided eight rebounds and six assists while Westbrook added 30 points, 12 rebounds and seven assists for the Thunder.
Denver was led by Ty Lawson with 29 points with eight assists.
J.J. Hickson notched 18 points with 19 rebounds.