Dawes had the only goal in the shootout, and Henrik Lundqvist turned aside Jamie Langenbrunner, Brian Gionta and Zach Parise in the tiebreaker after making 17 saves through overtime.
The only blemish was Dainius Zubrus' power-play goal in the first period that looked as though it would be enough for New Jersey to finally top the Rangers.
Martin Brodeur faced only mild pressure from the Rangers, who rarely got quality scoring chances. Many drives came from sharp angles or were blocked well before they could get in on goal, until Drury scored.
Just 18 seconds after the Rangers' fourth failed power play expired, Drury tied it. Ryan Callahan swooped out from behind the net and fed a pretty left-to-right pass that Drury buried behind Brodeur with 4:46 left.
New Jersey leads Montreal by one point in the race for the top playoff seed in the East. The Devils are four points ahead of the Rangers in the Atlantic Division.
Working solely in the Rangers' zone during their first power play of the night, the Devils grabbed the lead at 6:05 of the first. Mike Rupp played the puck behind the New York net and fed a quick pass in front to Zubrus, who cut from the right wing boards between a pair of Rangers and one-timed a shot by Lundqvist.
That marked just the second time in six games with the Rangers this season that the Devils held a lead.
