With 1:34 left in the second quarter, on a 3-pointer by Jae Crowder, Paul notched career assist No. 10,336 to push past Suns legend Steve Nash for sole possession of third place.
Paul finished the night with a season-high 18 assists in the Suns' 112-100 win to give him 10,346 for his career. He now trails only John Stockton (15,806) and Jason Kidd (12,091) on the all-time list.
"I'm grateful and I thank God for this opportunity," Paul said. "I say it all the time, not only to be on a team and still be in the league but to still be playing. You don't take that for granted."
The 36-year-old Paul acknowledged he is unlikely to make it to the top of that leaderboard, however.
"I looked at that list when I first came into the league and seeing that John Stockton and I was like, 'I'm going to get it.' But that ship sailed. That ship sailed. It sailed a while ago when I started missing all them games," Paul said with a laugh. "But to be third and still be playing, I'm grateful for it and I still got a ways to go."
Paul also passed Mark Jackson with his seventh assist of the night, his final one of the first quarter. Paul started the night six assists behind Jackson (10,334) and seven behind Nash (10,335).
Earlier this season, Paul became the first player in NBA history with 20,000 career points and 10,000 career assists.
Paul also had 14 points Tuesday night, 10 of which came in the fourth quarter as the Suns outscored New Orleans 32-16 to secure the victory.
"We keep acting like this is normal," Suns guard Devin Booker said of Paul's achievements. "We go in the locker room like, 'Oh you passed Steve Nash, Mark Jackson. Game ball.' It's bigger than that, man. It really is. Obviously I know he was going to be close coming into today and he passed it by a lot. I'm just happy to be a part of it."
The assist that moved Paul up to No. 3 came against his original team, although it was the New Orleans Hornets back when he was drafted there in 2005.
In his first six seasons with New Orleans, Paul had 4,228 assists and led the league in assists per game in 2007-08 and 2008-09. From there, he moved on to the LA Clippers, where he totaled 4,023 assists in six seasons and again was a two-time league leader in assists per game in 2013-14 and 2014-15.
Paul's 9.4 assists per game average is fourth all time in the NBA behind Magic Johnson (11.2), Stockton (10.5) and Oscar Robertson (9.5).