Randolph had 15 points and 10 rebounds and the Grizzlies rode a strong second quarter to win in both teams' first game since the All Star break.
The Grizzlies gave up 30 points in the opening period, but allowed the same total in the second and third quarters combined.
"From there, we dominated the game," coach Lionel Hollins said. "We were in complete control."
The Grizzlies trailed by as many as 11 points in the first half, but ended it on a 23-3 run to lead 53-42 at intermission. The Pistons managed just one basket in the final eight minutes of the half and had twice as many turnovers as field goals (8-4) in the second quarter.
Memphis outscored the Pistons 31-12 in the period and never looked back.
Quincy Pondexter, who was whistled for a technical foul midway through the period, had all eight first-half points during the decisive run. He and Tony Allen each finished with 10 points and were a big part of the second-quarter surge.
"We were a little rusty," Allen said after the Grizzlies' first game in six days. "Once we got our composure we were able to do what we do best. We played great defense and pounded the ball inside."
Detroit's energy seemed to disappear in the second half. The lead passed the 20-point plateau after a 13-0 Memphis run midway through the third quarter and stayed there until the final minutes.
"Once we lost the lead they became the instigators and aggressors," Pistons coach Lawrence Frank said. "To me, Tony Allen and Quincy Pondexter changed the game."
Tayshaun Prince played his first game at the Palace of Auburn Hills since a Jan. 30 trade sent him and Austin Day to Memphis. The starter on Detroit's 2004 NBA championship team drew a loud ovation from the crowd of 13,481.
"I wish I would've played a little better," he said after scoring four points. "You always want to play well against your old team."
Official David Guthrie called technical fouls on Hollins and Pondexter, as well as Detroit G Will Bynum, during a 2-minute stretch of the second quarter. . The teams observed a moment of silence for Los Angeles Lakers owner Jerry Buss, who died Monday.