Erik Karlsson and Timo Meir each had a goal and two assists for the Sharks. Kevin Labanc also scored.
Adin Hill stopped all 21 shots he faced for the Sharks, who won their second straight to start the season and have won 10 straight overall against Montreal.
Canadiens goalie Jake Allen stopped 20 of 25 shots as Montreal - coming off a run to the Stanley Cup Final - lost its fourth straight to start a season for the first time since 1995.
''It's a trap when you get this close and you have to start again quickly,'' Canadiens coach Dominique Ducharme said. ''There is a mental aspect to this and right now we're not dealing with it well. But you need to turn the page, put on your working boots and work the right way.
''You can't just work, you need to work the right way. Right now, we're disconnected with and without the puck.''
Dahlen opened the scoring just 1:22 into the first period, working with Meir to redirect Brent Burns' shot from the blue line.
Dahlen added his second goal on a counterattack just over a minute later. Logan Couture found the Swede on a 2-on-1 and doubled the Sharks' lead.
''It feels pretty unrealistic right now, so maybe I need some time to let it sink in,'' Dahlen said. ''I felt that it hit my stick but I wasn't sure and everything went so quick. Then the second one just pops on my stick and I shoot. It was a pretty unreal three minutes.''
Dahlen called his father after the game.
''My phone plan didn't have Canada, so I had to find some Wi-Fi,'' Dahlen said. ''I've been calling him after almost every game since I was 10, so I gave him a call after this one too and it was pretty fun.''
Karlsson scored San Jose's third goal just before the first intermission. His shot from the blue line deflected onto David Savard and past Allen.
On the power play, Meier accepted Couture's pass and beat Allen to put San Jose 4-0 at 3:09 of the second period.
Sharks coach Bob Boughner is confident in the chemistry between Dahlen, Meier and Couture.
''We've kept them together for the most part and I think it's a combination. Dahlen is a real smart player, he's a big compliment to that line,'' Boughner said. ''And I think Timo found a different gear, he looked like a man among boys tonight.''
Labanc scored with about 5 1/2 minutes left in the game, his first of the season.
The Canadiens played out the final minutes under a flurry of boos at Bell Centre.
''Any time you go into these losing skids, it might be escalated in a market like Montreal where you need to focus on it wherever you go, it's hard to get out of these things,'' winger Brendan Gallagher said. ''It's on your mind all the time, it's all you think about, it's all you worry about. At times you try to do a little bit too much.''
Sharks: At Ottawa on Thursday.
Canadiens: Host Carolina Hurricanes on Thursday.
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