The normally high scoring Warriors made only four free throws for the entire game, had a four-game winning streak snapped and lost at home for the first time this year.
Neither side could produce a point during the final two minutes of the fourth period, leaving the game tied at 75-75. There were seven missed shots and two turnovers during that stretch with the Warriors' Andre Iguodala misfiring on a jumper at the buzzer.
It was a 79-79 game when Conley made his 3-pointer with 1:37 to play in the extra period. Tayshaun Prince then hit a pair of jumpers for Memphis to protect the lead.
The Grizzlies won all four games on a road trip that concluded with Wednesday's outing.
Golden State got 21 points from Klay Thompson and 18 from David Lee.