Blake Comeau tied it 3-3 at 9:08 of the third.
Mattias Janmark scored, and Joe Pavelski had two assists for Dallas (34-19-5), which is 6-1-1 in its past eight games and trails the St. Louis Blues by one point for first in the Western Conference and Central Division. Ben Bishop made 29 saves.
Montreal (27-26-8) has lost four straight, including a 4-1 loss at the Pittsburgh Penguins on Friday, since going 9-3-0 in its previous 12. The Canadiens are nine points out of the second wild card into the playoffs from the Eastern Conference.
"I think they just grabbed a lot of momentum and that carried on in the game," said Canadiens goalie Carey Price, who made 22 saves. "I thought we played a very excellent third period and deserved a better third than that."
It was the Stars' NHL-leading ninth win when trailing after the second period.
"We've been doing it all season, so I don't think there's any doubt in this room whenever we get down," Bishop said. "It's kind of the way we play and always stick to our systems. They got a couple of lucky bounces there to get their goals, and I don't think anybody was too worried. We were playing good. Obviously, they came at us pretty hot there at the beginning, but I thought we did a good job the rest of the game."
Joel Armia, Jordan Weal and Nick Cousins scored to give the Canadiens a 3-0 lead.
Armia made it 1-0 at 1:02 of the first, scoring off a rebound from the edge of the crease.