The Spurs took a six-point lead on Tony Parker's field goal with 3:57 to play but managed only two more points the rest of the way. Their last points came on two free throws by Tiago Splitter with 2:31 that gave them an 86-82 advantage.
San Antonio had won its first nine home games of the season. The Spurs are 1-5 on the road.
Sacramento has won two straight for the first time this season.
Splitter then missed a hook shot with 41 seconds left and the Spurs chose to play defense for the entire 24 seconds on the shot clock rather than foul.
Evans led the Kings with 23 points while both Marcus Thornton and DeMarcus Cousins had had 17. Parker paced the Spurs with 24 points.