Dawes added another with 3:53 left for his first career two-goal game.
It was almost the third straight 1-0 decision for the Rangers. Lundqvist, who has started all nine games this season, earned a shutout Saturday in a shootout loss at Boston and took a hard defeat in Pittsburgh on Tuesday.
Scott Gomez didn't get on the scoresheet in his first game against the Devils after seven seasons with New Jersey, but stressed earlier Thursday the importance of getting the win.
The Rangers didn't give their fans a chance to get frustrated at the start of a six-game homestand after a 0-2-1 road trip that produced only three goals, including their first back-to-back 1-0 losses.
Dawes struck just 57 seconds in after a poor decision by former Rangers defenceman Karel Rachunek, who sent the puck toward his bench while New Jersey changed defencemen. Andy Greene jumped on, then hopped over the boards when the puck arrived as Johnny Oduya came off.
To avoid a penalty for too many men, neither player went for the puck. That allowed Brendan Shanahan to corral it and send a pass to Dawes, who beat Martin Brodeur with a wrist shot for his first of the season and second in 12 career NHL games.
That snapped the Rangers' drought at 126 minutes, dating to last Thursday's 5-3 loss at Atlanta. It was also their first tally at even strength in 182:06, stretching to a 3-1 home loss against Ottawa on Oct. 13 that kicked off the skid.
Dawes snapped a shot from the centre of the right circle in the closing minutes to seal New York's much-needed win.
Brodeur, playing for the first time in a week after a rare game off, was sharp in making 29 saves. The Devils finished their season-starting, nine-game road trip 3-5-1 - including three shutout losses - and will open their new arena in Newark, N.J., on Saturday against Ottawa.
New York (3-5-1) has scored only 14 goals in nine games.
The Devils nearly got even in the second period when Travis Zajac deflected a shot off the right post during a delayed penalty call. The puck caromed off the pipe, hit the back of Lundqvist's right leg, and fell behind him - tantalizingly close to the goal line before it was swept away by defenceman Michal Rozsival.
New Jersey failed on two power-play chances in the second period, but nearly yielded a short-handed goal when Marcel Hossa sent Chris Drury on a breakaway. His backhander was knocked aside by Brodeur.
Notes: The Rangers have never finished a homestand of at least six games under .500. ... Brodeur has played in 69 of 71 Devils games against the Rangers, dating to Dec. 26, 1993. ... New York D Thomas Pock cleared waivers and was assigned to AHL Hartford. ... Detroit in November 1954 was the last NHL team to play three consecutive 1-0 games, according to Elias Sports Bureau,