Utah went into the contest needing a victory over Memphis and a loss by the Los Angeles Lakers against Houston in order to earn the No. 8 seed in the Western Conference.
Those dreams began to fade in the third period, during which Memphis outscored the Jazz 25-13.
Randolph scored 25 points and grabbed 19 rebounds, boosting Memphis to its second straight win. Memphis posted a 56-26 regular-season record and will take on the Los Angeles Clippers in the first round of the playoffs.
Al Jefferson had 22 points for Utah, which at 43-39 finished with the best record among the 12 NBA teams that will not take part in the postseason.
The Lakers were able to start their home game late Wednesday knowing it had the No. 8 playoff berth secured.