David Clarkson tallied the only goal for the Devils, who have now won five straight.
The Rangers appeared to tie the game with 3.5 seconds left, but Marian Gaborik crashed into Brodeur and was called for goalie interference, negating Artem Anisimov's potential equalizer.
"The difference was Brodeur," said Rangers head coach John Tortorella, who refused to comment on the controversial ending. "He made some terrific saves in the third period."
Henrik Lundqvist was again marvelous in net, stopping 21-of-22 shots, but saw a personal four-game winning streak come to an end. New York, meanwhile, lost for only the second time in its last six games overall.
Brodeur, though, was the story, as the four-time Vezina Trophy winner stopped 30 shots, including 15 in the final period, to secure his first shutout since blanking the Pittsburgh Penguins on March 25 of last season.
It looked as if he would be denied the shutout late. With the Rangers net empty and time ticking down, Ryan Callahan wristed a shot from the left circle that Brodeur pushed aside. Gaborik then with a little help from Devils defenseman Anton Volchenkov went into Brodeur, knocking him backwards into the net, while Anisimov buried the putback, only to see it waved off by the penalty.
"I don't understand the call. If I was running I wouldn't say a word but I tried to stop, put my stick for Cally's pass and he pushes me into Brodeur," said Gaborik. "It's frustrating. We should be in overtime right now."
The Rangers had their chances in the third period and outshot the Devils, 15-1. New York was also 0-for-3 on the power play and is scoreless in its last 17 tries with the extra skater.
The Devils took advantage of a Stu Bickel holding call and cashed in on the power play 8:14 into the opening period. From behind the net Zach Parise found a streaking Clarkson, who beat Lundqvist glove side for his 21st goal of the season.
Clarkson now has five goals in his last five games, while Parise extended his point streak to six games (5g, 3a).
Petr Sykora almost made it a two-goal game late in the first period, but his rocket from the top of the right circle hit the crossbar with time winding down.
The Rangers had a pair of terrific chances in the second period. Shortly into the period Brandon Prust missed the net, while with time ticking down in the stanza Brodeur used the left glove to turn aside a terrific opportunity from Carl Hagelin.
Lundqvist has allowed one goal or fewer in 11 of his last 19 games, and has held opponents to two or fewer goals in 27 games this season...The Devils have outscored the opposition 21-12 during their five game win streak.