"I just wanted to get up the floor as quick as possible and find a shot," said Durant, who finished second in league MVP voting announced Sunday. "That was the only shot I could find and, by the grace of God, it went in."
Game 2 is Tuesday night in Oklahoma City.
Thabo Sefolosha kept the Grizzlies from going back ahead, deflecting a pass that Conley was able to corral but only after diving out of bounds. Reggie Jackson then hit a pair of free throws for a three-point lead, and Quincy Pondexter couldn't force overtime after getting fouled while attempting a 3-pointer with 1.6 seconds left.
Pondexter, a 72 percent career free-throw shooter, missed the first free throw. He made his second attempt before intentionally missing the third, but Durant swatted the rebound away and Marc Gasol's attempt at a buzzer-beater was late.
"We couldn't get stops. That's why we lost," said Gasol, who had 20 points and 10 rebounds. "We didn't lose because of free throws."
Kevin Martin scored 25 for Oklahoma City, which trailed for much of the game but was able to avoid repeating its Game 1 loss from when these two teams met in the West semifinals two years ago. The Thunder were able to rally and win that series in seven.
Fisher, a five-time NBA champion who joined the Thunder only after asking to be released by Dallas for family reasons earlier in the season, turned this game around with his defense.
Conley had gotten past him on a drive to the basket when Fisher reached in from behind and knocked the ball free. It was just the stop Oklahoma City needed.
"He makes plays, whether it's tipping the ball out and getting a steal or hitting the big shot to start the fourth," Durant said. "Or just his leadership in the huddles and the locker room is big for us. We need that."
Zach Randolph chipped in 18 points and 10 rebounds for the Grizzlies, and Pondexter and Conley scored 13 apiece. As a team, Memphis went 14 for 24 on free throws. Oklahoma City, which was the league's top foul shooting team with the third-best mark in NBA history, was 22 of 25.
"Obviously, we have to make free throws, especially myself," said Pondexter, who was slapped on his right arm by Jackson on his attempt at the tying 3-pointer. "We've just got to take this as a learning experience and move on."
After letting a seven-point lead slip away in the first half, Memphis surged back ahead with a 15-4 run early in the third quarter that featured two driving layups and a jumper by Conley. The lead stretched to 70-58 when Conley hit a free throw following 3-pointers on back-to-back possessions by Pondexter and Tayshaun Prince.
The Thunder started to rally before Pondexter's buzzer-beater from the half-court logo to finish the third quarter bumped the lead up to 73-64. That didn't stop Oklahoma City's comeback, though.
Fisher opened the final period with a 3-pointer and Martin converted a three-point play and a 3-pointer as the Thunder went on a 9-2 run.
"That shot at the buzzer, that's a great shot on his part. That's one of those things that it could have really took us back a little," Thunder coach Scott Brooks said, "but we still fought through that shot and kept competing when we were down."