After Kovalchuk's shootout tally, Zach Parise tacked one on for the Devils in the second round and New Jersey goalie Johan Hedberg stymied the Predators' Cal O'Reilly and Matt Halischuk to wrap up the win.
Nashville collapsed after taking a two-goal lead into the third period, allowing goal by Kovalchuk and David Clarkson to put the game in overtime.
Hedberg, playing in place of the injured Martin Brodeur, made 29 saves.
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