Drury redirected Nigel Dawes' shot past Montreal goalie Cristobal Huet at the right edge of the crease as New York overcame a 3-0 deficit early in the second period to extend its winning streak to three.
"Hopefully, it says a lot," Drury said. "Hopefully, we grab on to this and it carries us to some more wins down the stretch here because that was definitely a big win for us, on the road and down by three."
Martin Straka capped the five-goal rally at 18:30 of the third with the Rangers' second goal of the period.
"This was a huge character win," Rangers coach Tom Renney said. "If anything punctuates our last three games, this is it. If anything punctuates the character of our team, this is it, but every time we go to the rink moving forward it's going to check our character and we have to make sure that we use this as a reference point. It's our job to make sure this is a turning point in our season. It's that simple."
After Montreal rookie Sergei Kostitsyn beat Henrik Lundqvist on a penalty shot 5:40 into the second to put the Canadiens up by three, New York scored three times to close out the period. Michal Rozsival and Brandon Dubinsky scored 24 seconds apart to draw the Rangers to 3-2. Scott Gomez scored at 15:40 to tie it at 3.
Andrei Markov and Alex Kovalev each scored power-play goals for Montreal in the first period.
Lundqvist made 25 saves, including a huge stop 55 seconds into the third when he stuck out his right pad to deny Tomas Plekanec after the Canadiens center deked past him on a short-handed breakaway opportunity.
"Sometimes it's tough when they come from the side with a lot of speed," Lundqvist said. "He came fast so I tried to poke-check him and he went around, so I tried to be solid with my pad and it worked."
Huet also stopped 25 shots in his 12th straight start for Montreal, which lost an opportunity to move within one point of Eastern Conference leaders Ottawa for the Northeast Division lead.
"It was tough," Canadiens defenseman Mike Komisarek said. "We battled so hard for the first period. We talked about having a great start, we talked about dictating the level of play and having them adjust to us, and we did exactly that but we're definitely not going to win the game in the first 20 minutes. They got better and better as the game went on."
The Canadiens, who had won eight of 11 (8-2-1), host the Senators on Tuesday night.
Rangers left wing Ryan Hollweg got a boarding major and a game misconduct 15:40 into the second. Hollweg, who was leveled by Kovalev's shoulder check earlier in the period, was sent off after he checked Kostitsyn from behind headfirst into the boards.
"It was a dirty hit," said Canadiens defenseman Josh Gorges, who got a roughing minor for going after Hollweg immediately after the hit. "I think if you ask anybody that'll watch it, (Kostitsyn) had his back turned and there was no way to defend himself. Those kind of things we don't need in our game. That's how players get hurt."
Markov extended his points streak to six games when he opened the scoring 3:11 in with an unassisted effort for his 12th goal of the season.
Kovalev scored his team-leading 24th goal at 13:36 when he finished off a passing play with Saku Koivu and Chris Higgins to put Montreal up 2-0.
Kostitsyn made the most of an opportunity to increase the lead to 3-0 when he was awarded a penalty shot by referee Chris Lee after he was hooked by Sean Avery on a breakaway
Roszival put New York on the board with a power-play goal at 8:39 and Dubinsky scored on an unassisted effort at 9:03 to draw the Rangers to 3-2.
