The Jazz quickly fouled, but Matt Barnes made two free throws with 17.5 seconds left to doom Utah, which lost its first home game after six straight victories.
Blake Griffin finished with 30 points for the Clippers.
Mo Williams topped Utah with 20 points.
The Jazz and Clippers were tied, 15-15, with 4:10 left in the first quarter when Jefferson scored and triggered a 16-4 run.
Los Angeles made only one of its final 10 shots, while Utah ended the quarter shooting 74 percent.
The Clippers warmed up in the second quarter and the Jazz could extend their lead.
Jamal Crawford scored nine points and Griffin contributed seven in the final four minutes as L.A. narrowed what had been a 56-46 deficit.
Gordon Hayward scored 13 points and Williams added nine to help stake the Jazz to its 58-51 halftime lead.
Griffin led L.A. with 15 points.