A victory in a rematch with Utah on Sunday in Los Angeles would give the Clippers a perfect month of December. Their last loss came on Nov. 26 against New Orleans.
Paul scored the final seven Los Angeles points and finished with 29. His 3-point play with 17 seconds to go put the Clippers in front by two, but Al Jefferson made two foul shots for the Jazz to again tie the game.
After Paul hit his two free throws to give the Clippers the lead, Randy Foye missed a 3-point try that would have brought Utah the win.
The Los Angeles winning streak is the longest in club history and also the longest in the NBA this season. The Clippers' 24-6 record is tops in the league.
Blake Griffin hit 7-of-10 shots from the field to score 22 points for Los Angeles and was one of six Clippers in double figures.
Utah, which lost at home for the fourth time this year, was paced by Foye's 28 points.