Thornton and Cousins, who had 18 rebounds in the victory, were among five Sacramento players in double figures and compensated for a woeful 25-percent shooting percentage from 3-point range.
The win was the second in a row for the Kings following a six-game losing streak.
Devin Harris led Utah with 18 points. Josh Howard scored 15 in the fourth straight loss for the Jazz.
Utah led 46-44 at halftime, but the Kings scored 30 points in the third quarter and then got five clutch free throws from Thornton in the final 18 seconds to keep the game out of reach.