That was enough to rally the Rangers, who trailed 2-1 entering the third period, and send the Devils to their fifth straight loss (0-4-1). The teams are tied in the Eastern Conference playoff race with 91 points, but New Jersey is still ahead because it owns more wins than New York.
The teams have five games remaining, including one more head-to-head matchup on the final day of the season.
Christian Backman made up for his second-period gaffe that led to the Devils' go-ahead goal by tying it in the third during a power play. Brendan Shanahan had given the Rangers a 1-0 lead in the first by helping New York snap out of a drought with the man-advantage.
Former Devils star Scott Gomez shook off the effects of sore ribs and had two assists following a one-game absence for the Rangers. Henrik Lundqvist stopped 32 saves for his 35th win.
Brodeur made 28 saves, including a snatching glove shot during Ryan Callahan's third-period breakaway.
Patrik Elias and Zach Parise scored in the second for the Devils, on their longest losing streak since a six-game skid in November 2000.
Backman set up Dawes' heroics when he took a pass from Gomez a few feet inside the blue line and snapped a shot that beat Brodeur stickside at 1:38 of the third.
Parise's 31st goal was made possible by Backman's turnover on a clearing attempt. The Rangers defenseman skated backward in the zone with the puck, but instead of going up the boards to get it out, he put it into open ice and right on Parise's stick.
The speedy forward was in alone on Lundqvist with a few quick strides and ripped a drive in to put the Devils ahead 2-1 with 1:47 left in the second period. New Jersey hadn't scored as many as two goals in its four-game skid, managing a total of three.
Elias had tied it about 7 minutes earlier when he deflected a right-point shot from defenseman Paul Martin past Lundqvist at 11:38. That snapped the Devils' scoreless streak at 142 minutes, 34 seconds - mostly accrued during a home-and-home sweep by Pittsburgh in which New Jersey was outscored 9-1.
