John Madden picked up his first of the season, while Zach Parise extended his point streak to three games with an assist.
Martin Brodeur posted 20 saves in the defeat and remains 11 games behind Patrick Roy for the all-time wins record (551).
The loss extended the Devils' troubles against the Rangers (5-0-0), who took seven of eight during last year's regular season and captured the first-round playoff series between the teams in five games.
Since the 2005 lockout, MSG has been less than inviting for Jersey's Team. The Devils (2-1-0) have won just three times in their last 13 trips to Broadway since '05-06.
It marks a reversal of the Devils' past dominance on Rangers' home ice, where in the seven seasons from 1997-98 to 2003-04, New Jersey lost just once in 18 visits. The Devils last win at the Garden came in a 3-2 shootout on Feb. 22, 2007.
After Madden had cut the deficit to 2-1 late in the second period, Johnny Oduya was whistled for interfering with Ryan Callahan in front of the Devils' net. With 2.9 seconds left in the middle frame, the penalty time carried over 1:57 into the third, and Voros converted for his second of the night.
Voros' tip-in off of Paul Mara's shot made it 3-1 just 1:39 into the third and cemented the New York lead.
The Devils went 0-for-3 with the man advantage, and have misfired on their last nine power-play chances. That, commented head coach Brent Sutter, is where Monday's game was lost.
"Specialty teams were the difference," Sutter said. "We took a bad penalty at the end of the second, and started the third period by scoring on the power play. That was really the shift in the momentum. I thought we were carrying the play pretty well in the second. The last 12 minutes of the first they came at us pretty good. We had too many turnovers and we were forcing things. Then we settled down and played a real good second period. But, like I said, they got two power play goals and our power play didn't produce. That's the difference."
New Jersey got off on the wrong foot again Monday, allowing the first goal for a third straight game. The club has yet to hold a lead after the first period this season, and that trend continued after Brandon Dubinsky put the Rangers ahead 1-0.
When Brodeur attempted to catch Dubinsky's shoulder-high floater from outside the left circle, the puck dipped off Brodeur's thumb and into the net at 13:30 of the first.
New Jersey came up short on their lone power play chance of the first period, and were outshot 11-5 over the first 20 minutes. The Devils went the first six minutes of the middle frame before recording their sixth shot of the contest. Shortly afterward, Voros upped the Rangers' lead to 2-0 with his first of the game, and second of the season.
Voros, who spent much of the game screening Brodeur from the top of the crease, tipped Dan Girardi's shot top shelf on the power play to give the Rangers a two-goal advantage at 6:37 of the second.
Later in the middle frame, the Devils put the Rangers off balance thanks to a seven-minute flurry that was sparked by the checking line of Dainius Zubrus, Bobby Holik and Mike Rupp. Holik nearly cashed in with 6:48 remaining on a nifty feed from Zubrus behind the net, but was thwarted by Lundqvist.
On the next shift, the top line with Patrik Elias, Brian Rolston and Brian Gionta maintained the pressure as New Jersey turned the tables, outshooting New York 12-4 in the second.
Later, with David Clarkson in the trainer's room for repairs, Parise took Clarkson's place on the third line with Madden and Jay Pandolfo, and helped generate the Devils' goal.
Parise picked off a Dubinsky clearing attempt along the right boards before flinging a long shot attempt on Lundqvist. Madden found the loose puck through a crowd and backhanded his first of the season at 18:12 to trim the Rangers' lead to one.
Parise (2g-1a) has a point in three straight games, while Madden, who had a career-high 43 points in '07-08, became the third Devil to score a goal this season (Parise, Elias). It was the fifth goal in three games for New Jersey, who is still looking to get things going offensively.
"We haven't scored much," said Madden. "That troubled us a lot last year, but I think the difference is we were shutout 10 times last year. Fortunately, we haven't been shutout yet, but we've come pretty close. We have to find ways to score goals, whether it's dirty goals that come from mucking it up or getting rebounds. Three goals they got tonight weren't pretty: shots from the point with a tip and one that kind of fooled Marty. Those are goals that are going to go in, and some nights those are going to cost you. Tonight they did."
New York's Callahan added an empty-netter to complete the scoring with 58 seconds left in regulation.
After three straight contests against Atlantic opponents, the Devils play their first games outside the division when they travel to Atlanta on Thursday. They close out the four-game road trip at Washington on Saturday.
NJD NOTES Voros was originally selected with New Jersey's eighth pick in the 2001 Entry Draft... Elias' goal/point streak was snapped at two games... Sheldon Brookbank, Andy Greene and Petr Vrana were scratches.