Draisaitl had a goal and two assists, and McDavid had three assists for Edmonton (9-8-1), which ended a four-game losing streak. Mikko Koskinen made 27 saves.
"We went after our team pretty good and held them accountable (Monday) and it was a good response game," Oilers coach Todd McLellan said. "It shows me that they're attentive, their intent is strong and direct and good. They care about each other and that's a good sign. Taking that lesson and playing tonight's game is great, but the lesson has to arrive earlier so that the four-game losing streak is only a two-game losing streak or a three game. But we're still a team in evolution and we're learning lessons as we go. Stopped it at four instead of five, so that's positive."
WATCH: [All Canadiens vs. Oilers highlights]
Andrew Shaw had a goal and assist, and Max Domi scored for Montreal (9-6-3). Antti Niemi made 37 saves in his second straight start in place of Carey Price.
"It was called a no-show on our part," Canadiens coach Claude Julien said. "We are a fast skating team, so it wasn't about speed, but their team looked pretty fast when you give them the room that we did tonight. We didn't show up. It was a tough night, by far our worst game of the year. We have to, hopefully, write it off and move on."
Matt Benning scored his first goal of the season to give the Oilers a 1-0 lead at 3:13 of the first period when his point shot bounced off the end boards, hit the back of Niemi's leg and trickled into the net.