Gallinari and Faried combined to haul down 17 rebounds for Denver, which tied its franchise record for longest winning streak.
The Nuggets last won 15 consecutive games during the 1969-70 season, when the team was known as the Rockets and was a member of the pre-merger American Basketball Association.
Andre Iguodala helped out with 15 points with eight assists in Denver's 17th straight home victory.
DeMarcus Cousins had 24 points and 15 rebounds for the Kings, whose two-game winning streak was snapped.
Celebrity Families of 2014 [PHOTOS]