"This one has to hurt," Paul said. "If it doesn't hurt, it means you don't care."
Gasol scored 23 points, Randolph had 18 points and 16 rebounds, and Rudy Gay and Mike Conley added 13 points each to help the Grizzlies win for the first time in the Western Conference series at Staples Center and stave off elimination.
"We were very focused on our task defensively, and that's a big reason we won," said Gasol. "We were way more disciplined. The whole team is doing a better job of finding me in the post."
Griffin scored 17 points despite a sprained left knee that limited his jumping ability, and Eric Bledoe added 14 off the bench to lead the Clippers, who blew an eight-point lead in the fourth along with a second consecutive chance to close out what would have been a landmark playoff victory for the beleaguered franchise.
"Right now is not the time to start pointing the finger," Griffin said.
Paul scored 11 points playing with a strained right hip flexor and a jammed right middle finger.
"I didn't know he was hurt," Randolph said. "Everybody's hurting. My knee is hurting. I ain't looking for no excuse. You got to put that in the back and keep playing."
The Clippers are seeking just the third playoff series win in franchise history. Game 7 will be Sunday in Memphis.
"It's uncharted territory for us as a team," Clippers second-year coach Vinny Del Negro said. "We got to stay together, battle through it and give our best effort of the season."