Carter's first goal on a power play broke a scoreless tie in the second period, with his second coming 4:39 into the third period for a 2-0 Kings advantage.
Only 19 seconds later, the winger stole the puck off a face-off in the Predators' zone and fired another shot past Pekka Rinne for an unassisted tally and his fifth career hat trick.
Voynov assisted on two of Carter's scores as well as a Mike Richards tally as the resurgent Kings notched their sixth win in seven games.
Dwight King also scored and Jonathan Bernier turned aside 18-of-19 shots in the victory.
Rinne allowed five goals on 13 shots before leaving the net in the third period in favor of Chris Mason, who turned aside the final three Los Angeles shots.
Gabriel Bourque registered Nashville's only goal.