Noah Dobson had two assists, and Jakub Dobes made 24 saves for the Canadiens (9-3-2), who had won their first five games that ended in overtime this season.
Dobes had won his first six starts this season with a 1.97 goals-against average and .930 save percentage. However, the 24-year-old rookie goalie became emotional when speaking with the media postgame, blaming himself for the loss.
"I think we played good enough to win, but it's just not good enough from my side. I don't know, it's really hard right now," Dobes said. "I need to be better, 6-0 is not good enough."
Dobson defended his goalie.
"I think he's being hard on himself," he said. "I think he's played great hockey for us. He's a gamer. He keeps things light, which is kind of his personality. But when we get to the rink for game time and he's starting, he's a gamer, and you see that out there. His compete level on every play is high. We all have his back. He's been great for us all year. I'm sure he will continue to do so."
Jake Evans tied it 2-2 for Montreal just 59 seconds into the third period on a sharp-angled shot from the right face-off circle that trickled under Markstrom's left pad.
Oliver Kapanen then put Montreal ahead 3-2 at 10:33, converting on a rebound with a backhand from the slot.
"I think we played well tonight," Canadiens forward Kirby Dach said. "We did a lot of the right things and had a lot of chances. We just got to find a way to start closing those games out."
Meier tied it 3-3 with 1:07 remaining in regulation when he scored on a scramble in the crease at the right post.
"We showed character this game," Meier said. "Sometimes during the season there's going to be games like this where it's great when you find a way to win and it feels good, but you definitely got to know what you did right and what didn't go your way."