Much like the rivalry itself, the Lakers' early control dissolved into a tight affair before Kobe Bryant nailed a jumper to put his team four points ahead with 21 seconds left and his team held on.
Bryant had 31 points and big man Andrew Bynum returned from a one-game absence with an ankle injury to score 36.
"There's a lot of haters out there that forget the tradition of the Lakers," Bynum said.
Caron Butler tallied a team-high 28 points and Chris Paul got 22 and 16 assists but the Clippers had their six-game win streak snapped and fell 2 1/2 games behind the Lakers for third place in the West.
The Clippers trailed by as much as 15 in the second half but ignited a run that gave them a one-point lead late in the fourth before the Lakers fought back to regain supremacy.
In other NBA action, the streaking San Antonio Spurs edged out Boston 87-86 at the final buzzer to record a ninth straight victory that pulled them to within one game of West-leading Oklahoma City.
"The intensity on both teams had that playoff atmosphere. You could tell by the sense of urgency on both parts," James said after scoring 34 points to Durant's 30. "We were just happy we were able to come away with a win."
The Heat are now two games back of East-leading Chicago.
The defending NBA champion Dallas Mavericks topped Memphis 95-85 to pull even with the Grizzlies in the standings as they jostle for the West's fifth spot.