''It's taking a weight off my shoulders, I can tell you that personally,'' Sabres coach Don Granato said. ''It's fun to see the guys play that way, obviously.''
Tage Thompson added a goal and three assists, while Rasmus Dahlin and Alex Tuch each had a goal and two assists.
JJ Peterka and Henri Jokiharju also scored for Buffalo (8-11-0) in its first win since Nov. 2. Craig Anderson made 29 saves.
''They care and they work and we're young,'' Granato said. ''If I had an older team, I wouldn't be in these situations, and you go through the streaks. They know where they're at. These guys get better because of this stuff.''
Cole Caufield and Sean Monahan scored for Montreal (9-9-1). Jake Allen allowed seven goals on 38 shots.
''I think it was a bad game all around,'' defenseman David Savard said. ''We've got to move past it. And obviously we've got some stuff to learn, which is that bad start in the first period and it put us behind.
''It's a couple of games now and we've got to make sure we're ready tomorrow to go and play our game. We've been in every game so far if we play our style and our system, and we didn't do that tonight.''
The Canadiens went 0 for 6 on the power play, including a 4-on-3 opportunity halfway through the second period.
''I think the first two power plays had some good chances. The last four weren't very strong,'' coach Martin St. Louis said. ''We want to create scoring chances and I think apart from tonight, our power play in the last 10 games was still pretty good. It's just that tonight wasn't our night.''
Buffalo jumped out to a 3-0 lead in the first 2:13, the second-fastest three goals to begin a game in franchise history.
Thompson found Dahlin in the slot only 35 seconds in. Just 17 seconds later, the Sabres attacked Allen's crease and Peterka netted Buffalo's second of the night.
In an attempt to shift the momentum, Canadiens defenseman Chris Wideman dropped the gloves with Sabres forward Tyson Jost after Montreal's Michael Pezetta was called for boarding.
Instead, Tuch jumped on the opportunity to make it 3-0 with a wrist shot on the power play.
''I thought we were able to really help each other out, out there,'' Tuch said.
Montreal cut the deficit to two when Caufield sent a slap shot past Anderson at 2:57 of the first.
Buffalo regained its three-goal cushion 11:21 into the second when Thompson found Skinner in the slot with a backhand pass and the winger scored his sixth of the season.
Montreal made it 4-2 when Monahan took the puck from center ice, skated in and scored off his own rebound at 13:57.
The Sabres bumped their advantage to three again when Tuch rounded the net and sent a backhand pass to Jokiharju, who beat Allen for his first of the season at 3:48 of the third.
Thompson added his team's sixth goal at 6:39 with a wrist shot over Allen's glove.
Skinner iced it with a one-timer from the right faceoff circle at 14:26.
The 30-year-old Skinner has 40 points (23 goals, 17 assists) in 37 career games against the Canadiens.
''I think sometimes it's just sort of fortunate. It just kind of happens,'' he said. ''I think for me to have that kind of success, it usually means the team's playing well. I think our line played pretty well tonight so you just benefit from that.
COMING OF AGE
Dahlin is off to a strong start in his fifth NHL season. The top pick in the 2018 draft has 12 points (two goals, 10 assists) in his last 10 games.
Sabres: Host the St. Louis Blues on Wednesday.
Canadiens: At the Columbus Blue Jackets on Wednesday.
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