The Rangers scored twice in two minutes early in the period then sealed their win with a third goal just before the end of time after their cross-river rivals had dominated the first two periods but failed to capitalize on their chances.
New York defenseman Dan Girardi broke the deadlock when he scored on a slapshot 3:19 into the last period and rookie forward Chris Kreider gave the Rangers insurance with a tip-in goal at 5:16.
Rangers captain Ryan Callahan finished off the rout by scoring into an empty net when New Jersey pulled their goalie Martin Brodeur.
"I'm not sure how far we'll go from here or what will happen next but this is a team that stays with it," Rangers coach John Tortorella said.
"We still have things to improve on, but there's no panic, we know who we are and how we have to play."
The Devils, back on home ice after splitting the first two games at Madison Square Garden, outplayed the visitors in almost every aspect but could not find a way past New York's Swedish goalie Henrik Lundqvist, who made 36 saves and killed five penalties to register his second shutout of the series.
"I think their goalie was the difference," New Jersey head coach Peter DeBoer said.
"The tendency when you lose a game like this is to microscope everything but we played a real good hockey game but we lost because we didn't score a goal.
"If we had found a way to score in the first two periods it might have been a different but we can't feel sorry for ourselves, we just have to move on."
The Rangers, showing no signs of fatigue after being pushed to seven games in the two previous rounds of the NHL playoffs, seized their opportunity in a powerplay when New Jersey's Bryce Salvador was sent to the penalty box for hooking.
Brad Richards won a vital face-off and flicked the puck to the unmarked Girardi who sent it whistling past Brodeur.
The top-seeded Rangers scored again when Kreider, who was still playing college hockey when he was called up just before the playoffs, registered his fifth goal of the postseason, deflecting a shot from Ryan McDonagh.
The Devils struck the bar four minutes from the end in a last-ditch effort to get back into the game before Callahan compounded their frustrations with his empty net goal.
"We didn't panic and that's the big thing. We kept our composure," said Lundqvist.
"You need some luck some. We didn't play our best in the first half but I thought we were really good in the second half. It was a good day for us."
The teams return to Newark for Game Four on Monday with the eventual winner to play either Los Angeles or Phoenix for the Stanley Cup.