Memphis missed five free throws in the final 25 seconds, but still had enough of a lead to survive a 3-pointer in the closing seconds by the Clippers' Randy Foye that accounted for the final score.
The Grizzlies won their second straight game in the opening-round playoff series that will conclude Sunday in Memphis. The winner will meet top-seeded San Antonio in the Western Conference semifinals.
Los Angeles came from behind to take an eight-point lead early in the fourth quarter, but Memphis went on a 10-point run to regain the advantage.
Conley's 3-pointer boosted Memphis in front 83-80 and the Grizzlies held an 89-85 lead as the game entered its final half minute. Tony Allen then missed two free throws for Memphis and Zach Randolph missed 3-of-4, but the Clippers could not capitalize.
Los Angeles missed four field goal attempts in a row and turned the ball over twice down the stretch before Foye hit his meaningless 3-pointer.
Marc Gasol scored 23 points to lead the Grizzlies and Randolph had 18 points to go with 16 rebounds.