Rudolph put in 21 points and grabbed eight rebounds. Gay added 20.
Marc Gasol helped out with 10 points, 12 rebounds and five assists for the Grizzlies, who posted their third win in five games.
The Suns, meanwhile, continued to struggle, falling for the eighth time in nine contests as leading scorer Goran Dragic was held to nine points.
Phoenix got 17 points and five rebounds from P.J. Tucker.