Vesey, Nick Holden and Kevin Hayes scored, and Lundqvist made a season-high 45 saves for the Rangers (19-12-4). Mika Zibanejad scored in the shootout.
"We still found a way to get a point in a game where we didn't do that many good things, I think," Lundqvist said. "But you've got to give Jersey credit; the biggest thing is it felt like they wanted it more. We just couldn't match their intensity."
The Devils pulled into a 3-3 tie in the third period when Boyle scored a power-play goal off a backhand from the slot at 11:54.
"We showed maturity; the frustration is coming from a good spot," Boyle said. "You're allowed to be frustrated a little bit because you want to help the team, want to make a difference in the game. We're getting it from a bunch of different guys and it's important to pull the rope and contribute any way you can. As far as top to bottom in our lineup, everybody did that tonight."
Boyle, who played five seasons with the Rangers (2009-14), has seven points (three goals, four assists) in the past three games.
"The satisfaction is winning," Boyle said. "This rivalry wouldn't be what it is without both fan bases, and it's been like that for a number of years. We're out to prove we're legitimate, and we have to earn that every single night."