After Nashville's Craig Smith missed on the first try of the tiebreaker, Zibanejad sent a forehand shot past the blocker of Predators goalie Marek Mazanec.
Kyle Turris scored in regulation and Anderson finished with 32 saves for the Sens, who notched a fifth triumph in six games.
Mike Fisher got the only goal for the Predators, losers of five in six.
Mazanec turned aside 20-of-21 shots.
Notable deaths of 2014 [PHOTOS]
Arizona Cardinals player formally charged