Game 6 of the best-of-seven series is set for Thursday in Oklahoma City. A seventh game, if necessary, would be in San Antonio on Sunday.
San Antonio led 88-86 with 2:39 to play before Oklahoma City's Enes Kanter hit a layup, blocked a shot by the Spurs' Tim Duncan on the defensive end and ran the court to receive a pass from Westbrook for another layup that gave the Thunder a 90-88 lead.
San Antonio's Tony Parker hit a jumper to tie the game before Kevin Durant made two free throws to push Oklahoma City to a 92-90 advantage. Parker then went 1-for-2 from the line to bring the Spurs within 92-91, and, after a steal by Danny Green, badly missed a jumper with 11 seconds remaining.
Westbrook got loose on the ensuing inbounds play and headed to the basket as he was being fouled, converting the layup and a free throw with 6.3 seconds to play to ice the game.
Durant scored 23 points for the Thunder, while Steven Adams had 12 points and 11 rebounds. Kanter pulled down 13 rebounds for Oklahoma City, and Westbrook contributed 11 rebounds and nine assists.
The Spurs were led by Kawhi Leonard's 26 points. LaMarcus Aldridge and Green had 20 points apiece.
San Antonio, who lost just one game at home in the regular season, have now lost back-to-back contests in the AT&T Center to the Thunder. The Spurs had won seven consecutive Game 5s, with their last Game 5 loss coming in 2012, to Oklahoma City.
San Antonio looked out of sync from the start, falling into a 9-2 hole early in the first quarter. The Spurs fought back to take a 16-15 lead on a layup by Boris Diaw, but Oklahoma City scored the final seven points to reassume a 22-16 advantage after 12 minutes of play.
The Spurs grabbed a short-lived 32-31 lead on a Parker finger-roll layup early in the second quarter. Westbrook scored five straight points for Oklahoma City and Serge Ibaka poured in a wide-open 3-pointer to hand a 39-36 advantage back to the Thunder.
San Antonio ended the period with a 12-4 run that was punctuated by a ringing 3-pointer by Green with seven seconds remaining, and the Spurs owned a 48-43 advantage at halftime.
Leonard led all scorers with 17 first-half points, while Westbrook paced the Thunder with 14 points and Durant had 12 before the break.
Things went San Antonio's way early in the third quarter, as when Leonard dribbled the ball off his foot and it caromed straight to a wide-open Green, who grabbed the loose ball and rose in one motion to hit a 3-pointer that gave the Spurs a 59-47 lead.
Green hit another shot from beyond the arc on San Antonio's ensuing possession, and the Spurs had found their stride.
However, Oklahoma City were far from shaken by the San Antonio run, fashioning a 12-4 spurt of its own over the final four minutes to cut the Spurs' lead to 72-69 heading into the fourth quarter.
Green's fifth 3-pointer of the game with 8:55 to play was countered on ensuing possessions by consecutive treys by Westbrook and Durant that tied the game at 78 and set the stage for the furious finish.
