Trailing by eight points midway through the final quarter, the fourth-seeded Grizzlies fought back with an unbroken 10-0 run before regaining the lead for good when Mike Conley made a three-pointer with under five minutes remaining.
"We really didn't let their pressure bother us like we had in the previous games," an elated Conley said. "We had poise, poise down the stretch.
"They really do a great job of pressuring us but we got the ball, got it to our big guys and that's where we won the game.
"This is a game I know that we will all remember because our backs were against the wall. We had nothing to lose, we went out there and just played and now we have the opportunity to play Game Seven."
The series shifts back to Memphis for Game Seven on Sunday and the winners will go on to meet the top-seeded San Antonio Spurs in the Western Conference semi-finals.
Center Marc Gasol top-scored for Memphis with 23 points and forward Zach Randolph weighed in with 18 and 16 rebounds as they denied the heart-broken Clippers the chance to clinch a play-off series for only the third time, on their home court.
Power forward Blake Griffin led the way for Los Angeles with 17 points and Eric Bledsoe, off the bench, contributed 14 at a packed Staples Center but the home team paid the penalty after being out-rebounded 48-32 by their opponents.
The Clippers, who had led the series 3-1, came into Friday's game with injury concerns over their All-Star guard Chris Paul (a sore right hip and jammed middle finger) and Griffin (a sprained left knee). Neither player was at his best.
"Chris and Blake were battling through as best they could," Clippers coach Vinny Del Negro said. "They were trying to pull one out but we had to monitor their minutes a little bit."
Asked what had been the biggest difference in yet another tight game in a closely contested series, Del Negro replied: "The bottom line is their offensive rebounds. Zach and Marc hurt us in the post.
"We controlled the ball pretty well, we forced them into a bunch of turnovers, but we didn't shoot particularly well. We weren't able to convert a couple of times down the stretch ... we missed a few free throws."
The Grizzlies, who lost both regular-season games to the Clippers at the Staples Center along with the first two play-off games, made a fast start to lead 25-16 after the first quarter.
With Reggie Evans, Bledsoe and Kenyon Martin providing much needed energy off the bench for the Clippers, the home team then steadily clawed their way back to cut the deficit to 42-38 at halftime.
The Grizzlies increased their lead to eight points early in the third quarter before the Clippers again fought back, a Paul jump shot tying the score at 60-60 with just under four minutes remaining in the quarter.
Caron Butler, playing with a broken left hand, made a three-pointer to put Los Angeles ahead 63-62 before the teams entered the final quarter again deadlocked at 66-66.
An unbroken 10-0 run, punctuated by a Bledsoe jump shot, gave the Clippers a 76-68 lead with just over eight minutes remaining before the Grizzlies responded to level the score at 80-80.
Conley's three-pointer put Memphis ahead 83-80 and they never relinquished their grip as the Clippers missed several late scoring opportunities.