It was the Clippers' first win in Salt Lake City since 2003, snapping a string of 16 consecutive losses -- including a 108-79 loss in Utah on Jan. 17.
Paul sat out that the last visit with a left hamstring injury. He had 12 points in the fourth quarter in this game as the Clippers won their fourth straight.
Al Jefferson had 27 points and 12 rebounds to lead the Jazz.
The Clippers made eight free throws in the final 30 seconds.
Utah's C.J. Miles intentionally missed his second free throw with the Jazz down two points with 1.2 seconds left but Reggie Evans grabbed the rebound to seal the win.
The Jazz led by four points early in the fourth, but Paul and Mo Williams, who also missed the first meeting, brought the Clippers back.
|More on Clippers-Jazz|
|More NBA coverage|
Paul's 3-pointer gave Los Angeles an 86-84 lead with 6:08 remaining, and his reverse layup later made it 92-87.
Paul Millsap's 3-pointer got Utah to 101-99 with 16 seconds left, but Chauncey Billups made both free throws with 13 seconds remaining to put the Clippers up four.
Miles made a 3-pointer with 8 seconds left, but Billups made two more free throws to put Los Angeles up 105-102.
The Clippers fouled quickly again, and Devin Harris' free throws pulled Utah within one with 3.8 seconds left.
But Williams made both foul shots on the other end, and Miles was fouled bringing the ball up the court with 1.2 seconds left. He made one free throw before intentionally missing the second.
Overall, Utah struggled at the line, making 26 of 36 overall.
It was a back-and-forth fourth quarter after a high-flying first.
The game was under way just seven seconds in when Griffin took an alley-oop pass from Paul and dunked. He had five more dunks and 14 points before the quarter ended.
Paul and Griffin combined for 26 of L.A.'s 32 first-quarter points.
The Jazz countered with Jefferson, who had 10 points in the first quarter on 5-of-6 shooting.
Utah opened the second quarter on an 8-0 run to take a 37-32 lead, with Utah's under-21 players stealing the show. Rookie center Enes Kanter hit a pair of jump shots, second-year pro Derrick Favors followed with a driving layup and rookie Alec Burks scored inside on a feed from Earl Watson.
The Jazz extended their lead to 44-34, with Miles capping the 15-4 run with a three-point play.
Los Angeles answered with its own 8-0 run to pull to 44-42 midway through the quarter, fueled by Randy Foye.
Paul tied the score at 56 with a driving layup but Jefferson gave the Jazz a 58-56 halftime lead on his ninth field goal.
The Jazz were shooting 61 percent at halftime (27 of 44) while L.A. was at 53 percent (24 for 45).
The teams combined for only 32 points in the third quarter.
Clippers F Caron Butler did not play because of a sore lower back. Ryan Gomes started in his place but did not score. ... Mo Williams had a four-point play for the Clippers late in the first quarter after being fouled on a 3-point shot by Josh Howard. He finished with 19 points. ... Favors started the game 4 of 4 and had 10 points in the first half, and finished with 14. ... Billups and Paul were both assessed first-half technicals, while the officials called double-technicals on Gordon Hayward and Williams in the third. ... Jazz backup PG Watson left the game in the third quarter with a sprained left ankle. ... Jazz SG Raja Bell missed his second straight game with a strained adductor.