James and Dwyane Wade combined for 61 points as Miami beat the Indiana Pacers 95-86 but they lost forward Chris Bosh to an abdominal injury in the second quarter.
Chris Paul shook off an ailing hip to register 19 points for the Clippers, who reached the Western Conference semi-finals for the first time in six years.
They set up a second round showdown with the top-seeded San Antonio Spurs, starting in Texas on Tuesday.
Rudy Gay and Marc Gasol each scored a team-high 19 points for the Grizzlies who rallied from a 3-1 series deficit to force the winner-takes-all decider in Memphis.
The Grizzlies led by a point heading into the fourth quarter but Los Angeles went on an 11-2 run at the start of the period to seize control in a physical game.
"It's not what you wanted. It's not what you expected," said Memphis guard OJ Mayo. "But the results are the results. You've got to go into the off-season very motivated."
Bodies collided and players clashed during a tough contest that saw both teams shoot poorly but the Clippers find a way to prevail, helped by 41 points from their bench.
"That's what we do, the bench comes in and we try to change the pace of the game," reserve Eric Bledsoe told reporters. "We come in and get stops and just run."
In Miami, James hoisted the MVP trophy he was awarded on Saturday to the crowd before the game. Then, he lifted the Heat to victory, scoring 32 points and making 15 rebounds.
Wade added 29 for Miami who lost Bosh early. The Pacers led by six at halftime but failed to consolidate their advantage as the Heat assumed control.