Paul carried the Clippers in the final 3:46, scoring eight straight points, including a basket that gave them a 91-89 lead with 1:20 to play.
Mike Conley scored a career playoff high 28 points while leading the Grizzlies' fourth-quarter charge that came up just short. He had 10 points in the period, while Darrell Arthur scored five straight to pull Memphis into an 89-all tie.
Conley found Marc Gasol alone in the paint and the big man dunked to tie the game 91-all with 13 seconds left.
The Clippers won a hard-fought jump ball, but Paul missed a 3-pointer. He redeemed himself and set off a raucous reaction when he fought off Tony Allen's defense and banked in the game-winner.
Gasol added 17 points, Allen had 16 points and 10 rebounds, and Zach Randolph had 13 points while racking up five fouls for the second straight game.
The Clippers' bench started the fourth quarter and ran off eight straight points for the game's first double-digit lead, 81-71. Eric Bledsoe and Matt Barnes had four points each.
Conley answered with five straight points to close the Grizzlies within seven points.
Griffin and Paul joined the second unit, and Griffin scored on a driving dunk for an 85-76 lead. The Clippers returned to their high-flying ways after Lob City managed just one dunk in the series opener.
The Clippers led by seven points early in the third before the game turned into a back-and-forth affair. Memphis briefly regained a one-point lead and then tied it 59-all on Arthur's dunk before Paul scored six of the Clippers' final 13 points to send them into the fourth leading 75-71.
P Clippers at Grizzlies
Thursday, 7:30 p.m., TNT
Los Angeles leads series 2-0