Zach Randolph added 20 points and 10 rebounds for the Grizzlies, who will host Game 6 on Thursday.
Russell Westbrook had 30 points, 13 assists and 10 rebounds for the Thunder.
Oklahoma City trailed by 20 points in the third quarter but trimmed its deficit to six by the start of the fourth quarter and finally took a 79-78 lead on a 3-pointer by Durant.
Memphis maintained its composure. Consecutive baskets by Mike Conley put the Grizzlies up 87-82 with just under 4 minutes to play.
Oklahoma City's Caron Butler was fouled by Randolph as he hit a 3-pointer with 2:40 to play. He made the free throw to cut Memphis' lead to 87-86.
Oklahoma City, trailing 89-88, had a chance to take the lead in the final minute of regulation, but Durant was stripped, and the ball was stolen by Conley. Tony Allen made the first free throw with 30 seconds left. He missed the second, but Marc Gasol of the Grizzlies got the rebound with 20.3 seconds to play.
Conley was stripped by Westbrook, and Westbrook dunked with 4 seconds left to tie the game at 90. Memphis didn't get a shot off in time at the end of regulation and the teams went to overtime for a record fourth straight game.
NOTES: The Grizzlies wore their warmups inside out and the Thunder wore black socks and shoes in protest of Los Angeles Clippers owner Donald Sterling's incendiary racial remarks. ... The Grizzlies shot 60 percent from the field in the first quarter to lead 30-25. ... Oklahoma City made 6 of 9 3-point attempts in the first half, but just 9 of 34 shots inside the arc. ... Tulsa Shock guard Skylar Diggins attended the game.