Josh Anderson and Tyler Toffoli scored for Montreal (8-28-7), which has lost five straight (0-3-2) and is 1-7-4 in its past 12 games.
Sam Montembeault allowed six goals on 16 shots before being replaced at the start of the third period by Cayden Primeau, who made seven saves.
"We don't want any goalie going in net and having to make 50 saves a night," Toffoli said. "We're trying to figure those things out as players. They've been getting a lot of shots and doing a good job. We have to find a way to stop letting up Grade A opportunities. And I think they've been doing a good job of making big saves at the right time of games, and the games that we're in, they're keeping us in it."
Kane gave the Oilers a 1-0 lead at 11:21 of the first period when he tipped Evan Bouchard's point shot.
"It's great for him to jump in and get one early," Edmonton coach Dave Tippett said. "He looks in decent shape. You can tell he's rusty, he hadn't played in a while (since Dec. 18 with San Jose of the American Hockey League), so he'll have to get up to speed a little bit. But it's good to get him up and get him in the lineup and get him going."
It was the first time in 12 games that Edmonton scored first.
"It's nice to contribute on a new team coming in," Kane said. "And you want to show what you can do, and obviously it was great to get the boys on the board early, and everybody kind of took off after that."
Hyman redirected Kris Russell's point shot to make it 2-0 at 12:48, and Draisaitl scored 20 seconds later to extend the lead to 3-0.
Anderson got Montreal to within 3-1 at 3:58 of the second period before Ryan Nugent-Hopkins pushed the lead to 4-1 at 6:18.
Hyman made it 5-1 at 14:30 with a turnaround shot that beat Montembeault blocker side.
Toffoli cut it to 5-2 with a power-play goal at 15:09, three seconds after Zack Kassian was penalized for running into Montembeault behind the net. Toffoli scored on a one-timer from the left circle following a face-off win by Nick Suzuki.
Draisaitl scored 24 seconds later to increase the lead to 6-2. He has scored 31 goals this season, which is tied for the NHL lead with Chris Kreider of the New York Rangers.
"You've got to find that momentum and keep going, and I think we're struggling with that and then teams are pouncing on it," Canadiens forward Jake Evans said.