The Pelicans, who trailed the Clippers by one point, 54-53, at halftime, opened the third quarter with an 11-2 run and went on to take a 12-point lead into the fourth quarter, 88-76.
The Clippers went on a 12-0 run in the fourth quarter to cut the deficiit to one, 95-94, with just more than five minutes left. Paul scored 10 of those points over a two-minute period.
In the final minute of the game, the Pelicans retook control for a 106-100 lead. Jamal Crawford then made a quick three-pointer to cut the deficit to three, 106-103, with 22.5 seconds left, but with 17.9 seconds left, Jimmer Fredette made two free throws to give the Pelicans the win.
The Clippers outshot the Pelicans from the field, 45.7% to 41.9%. The Pelicans, however, made 13 of their 25 three-point attempts while the Clippers were only seven of 28 from beyond the arc. The Pelicans also outrebounded the Clippers, 51-38.
Paul led the Clippers with 24 points on 10-for-20 shooting, eight rebounds and seven assists. DeAndre Jordan had 12 points, 15 rebounds and three blocked shots.
J.J. Redick, who started Friday after missing Wednesday's game against the Utah Jazz because of back spasms, finished with 12 points and four assists. Austin Rivers scored in double digits for the Clippers for the second consecutive game, finishing with 10 points.
Pelicans All-Star power forward Anthony Davis did not play because of a left abductor strain. In his absence, former Clipper Eric Gordon had 28 points and seven assists while Ryan Anderson contributed 24 points and nine rebounds.
The loss bumped the Clippers (32-15) down to fifth place in the Western Conference. The Pelicans (25-22) are in ninth place. Before Friday's loss, the Clippers had won seven of their last eight against the Pelicans.
The Clippers will play their third game of an eight-game road trip against the San Antonio Spurs on Saturday.