Lundqvist was busiest in the second period, when he turned aside all 11 shots he saw en route to his fifth career post-season shutout.
He helped the Rangers grab a 1-0 lead in the best-of-seven series.
The game was scoreless until 53 seconds into the third period when Devils defenseman Bryce Salvador lost the puck at the blue line and Chris Kreider worked it to Dan Girardi at the left point.
The Rangers defenseman then unleashed a slap shot that beat Devils goaltender Martin Brodeur, providing all the offense New York would need.
Kreider himself scored 11 minutes later on a wrist shot off a feed from Artem Anisimov, and the latter added an empty-net tally to seal the Rangers' victory.
Brodeur finished with 25 saves for New Jersey.
Game 2 is set for Wednesday at Madison Square Garden.