Conley hit a 3-pointer from the left wing with 1:58 left to put the Grizzlies ahead to stay and spark a string of 10 straight Memphis points.
The Grizzlies head home to play three of the next four games, with Game 3 on Saturday in Memphis.
"Mike Conley is now one of the top five point guards in the league, whether anybody likes it or not," said Tony Allen, who had two of his five steals in the final minute.
"I know a lot of people have got their favorites on who they think it should be, but Mike Conley is in that conversation now, being able to do these types of things on the court night in and night out."
In the first round, Conley outdueled All-Star Chris Paul of the Los Angeles Clippers.
And now, he's filling the late-game void the Grizzlies created when they traded away leading scorer Rudy Gay in the middle of the season.
After Conley's go-ahead 3, he added an 18-foot jumper to stretch the lead to 94-90, then hit one of two free throws with 29.4 seconds left. He finished with 10 rebounds and nine assists, one shy of a triple-double.
"He played like he had been playing all season, and we needed that," coach Lionel Hollins said. "We needed to have somebody on the perimeter do something. He started getting to the basket a little bit and scored some big jump shots late."
After hitting the key baskets in Game 1, Durant couldn't provide an answer for the Thunder. He missed his last three shots, including a pair of 3-point attempts, and finished with 36 points, 11 rebounds and nine assists.
The Thunder caught a break when Allen tipped the ball away and Conley saved it from going out of bounds, only for it to end up in Durant's hands in the corner. But Durant was off-target on a 3-pointer, and Oklahoma City was forced to foul.
Zach Randolph added two free throws, and Allen then stole the ball from Durant and provided the finishing touches with a dunk. Derek Fisher hit a 3-pointer at the final buzzer for Oklahoma City.
Neither team led by more than seven in a game destined to come down to the wire.
Memphis piled up a 48-30 scoring advantage in the paint and also outscored Oklahoma City 23-6 in second-chance points and 18-7 on the fast break.
Storylines Late run keys win
R Mike Conley scores 26 points and Marc Gasol adds 24 for the Grizzlies.
• New York 105,Indiana 79
Series tied 1-1
• Memphis 99,Oklahoma City 93
Series tied 1-1
• Chicago at Miami,5 p.m., TNT
Bulls lead series 1-0
• Golden State at San Antonio, 7:30 p.m., TNT
Spurs lead series 1-0