Utah lost its 20th straight game on San Antonio's home floor.
The Spurs, who went into the contest trailing Houston by a half game in the battle for the Southwest Division title, won for just the third time in seven contests.
Utah was led by Deron Williams with 25 points and 10 assists. The Jazz, however, lost for the fifth time in six games and fell a game behind Dallas into eighth place in the Western Conference.
The eighth-place team in the West must take on the Los Angeles Lakers in the opening round of the playoffs.
San Antonio ended the third period on a 7-2 run to break a 69-69 tie.
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