Zuccarello's goal late in first period forged a 2-2 tie and he set up consecutive second-period scores by Chris Kreider and Benoit Pouliot to put New York ahead.
Lundqvist made 37 saves, including 15 in the third period, in notching his first regulation victory since Dec. 5.
Ondrej Palat, Tyler Johnson and Valtteri Filppula each scored for the Lightning.
Anders Lindback stopped 25-of-29 shots.
Celebrity Families of 2014 [PHOTOS]