Josh Anderson scored his third of the season and Brendan Gallagher added an empty-netter for the Canadiens, who had lost three of their previous four games - all on the road.
''It's been frustrating, obviously, but the only way to get through it is to keep working, keep grinding away,'' Anderson said. ''Once you do that, you're going to pull a lot of guys into the fight with you.''
Allen got his 23rd career shutout. Last Thursday, he made 45 saves against San Jose to get his first shutout with Montreal. He felt his teammates were more in control of this game.
''I thought it was our best game of the year,'' Allen said. ''We were solid all over the ice and it looked easy for us, to be honest, and that's the way it should be when we're all playing as one. I was really impressed with how they played.''
Alex Nedeljkovic finished with 39 saves for the Red Wings.
''They were better than us from the first shift all the way to the last shift,'' Detroit coach Jeff Blashill said. ''They played like it was a playoff game, we played like it was an exhibition game. That isn't even close to what we can do to be successful. We weren't even in the hemisphere of what we needed to do.
Montreal scored 2:06 after the opening puck drop as Suzuki got his first batting in his own rebound.
Anderson doubled the Canadiens' lead midway through the first period as he entered Detroit's zone from the left side all alone and beat Nedeljkovic with a his shot from the faceoff circle.
After allowing the two early goals, Nedeljkovic rallied in the second and third periods to give Detroit a chance at a comeback.
With Tyler Bertuzzi (unvaccinated) and Dylan Larkin (personal reasons) out of Detroit's lineup, the Red Wings couldn't solve Allen.
''Our execution wasn't good enough, our work ethic wasn't good enough,'' Blashill said. ''It was a 2-0 game, we had a chance to come out and have a great third period against a team that's just come off a West Coast trip, that lost a player early in Drouin and we didn't even come close to mustering up the effort.''
Gallagher's empty-netter sealed Montreal's second win of the season over Detroit.
Canadiens head coach Dominique Ducharme was satisfied with both the result and the way his team played.
''That's the way we play, that's our team,'' Ducharme said. ''That's what we need to do and it's the first time that we do that for 60 minutes. We saw glimpses of that but we need to do that for 60 minutes. It's not more complicated than that.''
Montreal forward Jonathan Drouin took a shot in the head from teammate Brett Kulak. Drouin hurried to the locker room and didn't return to the game. He was sent to the hospital for further evaluations.
''It's always worrisome but the news is encouraging,'' Ducharme said. ''He will get evaluated again tomorrow morning, but as of now, it's encouraging.''
Before the game, Allen was awarded the team's player-of-the-month award for October. He also had a shutout last Thursday against San Jose.
Red Wings: At Boston on Thursday night.
Canadiens: Host the New York Islanders on Thursday night in the second of a five-game homestand.
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