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.
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