Conley's trey with 30.7 seconds left forged a 90-90 tie and, after the Spurs' Tony Parker missed a jump shot, the Grizzlies' point guard sealed the win when he banked in the deciding score with sixth-tenths of a second remaining.
Conley ended with 23 points as Memphis posted a 12th straight home victory.
Jerryd Bayless came off the bench to collect 17 points for the Grizzlies, who trail the Southwest Division-leading Spurs by five games with eight to play.
Parker finished with 25 points for the Spurs, who were playing without the injured Manu Ginobili.
San Antonio's Tim Duncan and Kawhi Leonard each sat out the game to rest their sore left knees.
Turkey considering to use pistachios to heat country’s first eco-city
Moore to attend retreat in to avoid Kutcher's wedding