The victory left the Spurs a half game behind the Southwest Division-leading Charlotte Hornets. San Antonio has the third-best record in the Western Conference, one game ahead of Houston, Utah and Phoenix.
The Spurs played Monday without their leading scorer, Manu Ginobili, who missed his second straight game with a groin injury.
The Kings pulled to within 99-98 with 14.3 seconds left when Salmons completed a three-point play. The Kings fouled Finley on the next Spurs' possession and he sank two free throws to account for the winning margin.