A crowd of 50,105 witnessed the outdoor clash and it looked like the men from the Garden State were in for a good night as they rushed out into a 3-1 lead.
Patrick Elias had put the Devils ahead but Dominic Moore evened things up before Elias' second and a marker from Travis Zajac put New Jersey in control.
That was as good as it got for them, however, as Marc Staal reduced the deficit and Zuccarello's brace turned the game on its head.
Carl Hagelin and Rick Nash then extended the lead before Derek Stepan scored the first successful penalty shot in an outdoor NHL game.
Nick Bjugstad scored the only goal in the shootout as the Florida Panthers beat the Detroit Red Wings 5-4.
Sean Bergenheim had put the Panthers ahead but three unanswered goals from Gustav Nyquist, Daniel Alfredsson and Riley Sheahan put Detroit in control at the end of the second period.
Jesse Winchester and Tomas Tatar traded goals in the third and the Panthers took the game into a scoreless overtime period through Drew Shore and Brad Boyes.
Kevin Bieksa netted the overtime winner as the Vancouver Canucks overcame the Phoenix Coyotes by the same score.
Bieksa and Jannik Hansen had put Vancouver two up but two goals from Antoine Vermette levelled the scores.
Chris Higgins and Zac Dalpe then reopened the two-goal advantage but Vermette's hat-trick goal and a snipe from Shane Doan took the game to overtime.
The Winnipeg Jets produced a superb third-period performance to shock the Chicago Blackhawks 3-1.
Brandon Bollig gave the Stanley Cup champions the lead less than two minutes into the game but goals from Blake Wheeler and Andrew Ladd had Winnipeg ahead in the third before Wheeler's empty-netter sealed the win.
In Sunday's other game, the Edmonton Oilers claimed a 5-1 win over the Nashville Predators.
Justin Schultz and David Legwand traded goals before Jordan Eberle, Ryan Smyth, Nail Yakupov and Ryan Nugent-Hopkins took the game away from the Preds.