San Antonio shot 14 for 28 on 3-point attempts, two off its postseason record, including 5-for-7 shooting by Patty Mills.
Mills finished with 20 points and Tony Parker added 16.
Mike Conley had 26 points and Marc Gasol added 17 for the Grizzlies, who have lost nine straight postseason games in San Antonio.
With each team winning on its homecourt, Game 6 is Thursday night in Memphis.
