Paul's layup with 22 seconds left in regulation sent the game into overtime. He played 42 minutes, adding seven rebounds and going 12 of 13 at the free-throw line.
Denver had a chance to win in regulation, but JaVale McGee's 6-foot jumper from the right side was short at the buzzer.
Darren Collison had 27 points on 10-of-13 shooting for the Clippers (4-2), who outscored the Nuggets 14-7 in overtime.
Blake Griffin added 17 points and eight rebounds in 39 minutes for Los Angeles, which trailed 104-100 with 47 seconds left in regulation.
Paul had 15 points by halftime, when the Clippers trailed 47-46.
The Clippers shot 52 percent from the field and the Nuggets were at 49 percent.
This is the second straight year the Clippers played at the Mandalay Bay Events Center in Las Vegas. Last year, the first time the Clippers hosted a game in Las Vegas, Ty Lawson's layup at the buzzer gave Denver a 106-104 win.
The teams played the fourth game on the Las Vegas Strip.