Paul netted 27 points for the game, including a 16-foot jumper with 44 seconds left in the extra frame that put the Clippers ahead 99-93 and essentially sealed the victory.
The dynamic point guard grabbed nine rebounds and distributed seven assists for Los Angeles, which took a 3-1 lead over Memphis in their Western Conference quarterfinal series.
Blake Griffin added a game-high 30 points with seven assists and five boards before fouling out.
The Clippers were forced to go to overtime when the Grizzlies finished the fourth quarter on an 11-1 run, capped by Zach Randolph's free throws to tie the contest with 20 seconds left.
Mike Conley led the Grizzlies with 25 points and Rudy Gay finished with 23 for Memphis, who will attempt to avoid elimination in Wednesday's Game 5 at Memphis.