With the Clippers leading by just two points in the last minute, Paul twice pushed the Clippers' edge back to four with a jumper and two free throws and finished with his team's final five points to seal the road victory.
It was the seventh successive win for the Clippers (15-6), their longest stretch since taking eight straight in the 1991-92 campaign.
"Our team is amazing because it's a team effort every night," Paul said. "We don't have a guy who's going to score 30 or 40."
Paul finished with 18 points, Blake Griffin added 22 and 10 rebounds and Los Angeles also got 34 points from their bench.
Carlos Boozer led the way with 24 points and 13 rebounds for the Bulls, who fought back from a nine-point deficit in the fourth quarter to make the Clippers sweat.
Chicago (11-9), still playing without Derrick Rose as well as Richard Hamilton, had a three-game win streak snapped.
Trailing early, the Bulls surged ahead in the third where an 18-6 run gave them a 60-57 advantage.
Italian shooter Marco Belinelli added a three-pointer during the stretch and tallied 18 points but he made just six of 22 field goal attempts.
The Clippers turned the tables late in the third quarter as their reserves made an impact. Matt Barnes finished with 14 and Jamal Crawford also added 10 off the bench.
"This was a good road win for us, we've struggled on the road this season," Barnes said. "It took a little bit to get going, but we hit our stride."
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