After three rounds of the tiebreaker failed to produce a winner, Zibanejad drew Brodeur out of the net with a backhand fake and fired the puck high over his shoulder for the decisive score.
Ottawa's Ben Bishop followed with a glove save of Andrei Loktionov to wrap up the Senators' fifth victory in six games.
Colin Greening and Chris Phillips each scored and Bishop racked up 32 saves.
Loktionov and Marek Zidlicky provided the offense for New Jersey, which lost for the first time in three contests while trying to secure an Eastern Conference playoff spot.
Brodeur ended with just 12 saves.
2014: The Year in Music [PHOTOS]