Noah Juulsen made it 5-3 with his first NHL goal at 9:53 of the third.
"We're not playing well," New York coach Doug Weight said. "You can say what you want; we need more."
Galchenyuk completed the hat trick with an empty-net goal with 49 seconds remaining to make it 6-3.
"I think he's been playing well lately and he's had some chances," Canadiens coach Claude Julien said. "Tonight was one of those nights where there's an empty-net goal, an unbelievable setup by Reilly, but he can shoot the puck. When he's on top of his game, that's what he does best. It was nice to see him come out with a four-point night on his 400th (NHL) game. It looks good on him."