Randolph scored on a layup with 16 seconds to put the Grizzlies in front by three and he added two free throws with 9 seconds remaining for a five-point cushion.
Memphis handed the Lakers their third home loss in six games with Randolph scoring 28 points to go with 11 rebounds.
Marc Gasol flirted with a triple-double, scoring 18 points, grabbing eight rebounds and handing out eight assists.
A relatively low total of 30 fouls were called in the game and the teams combined to take only 25 free throws. Memphis made all 14 it attempted.