The game was deadlocked midway through the fourth period before the Grizzlies went on a late run that brought them their fourth straight win.
Memphis stayed within a half game of the Los Angeles Clippers for the No. 4 seed in the Western Conference and home-court advantage in the first round of the playoffs.
Phoenix dropped behind Utah into 10th place in the West and is two games out of a post-season spot.
Phoenix was led by Marcin Gortat with 19 points. The Suns had won five of their previous six.