Lucic made it 3-0 on the power play at 4:21 of the second period, finishing off a three-way passing play started with a cross-ice pass from McDavid to Draisaitl.
"The last two games, we had teams coming in on back-to-backs, and this year, we haven't been great in the first period, so we wanted to address that," Oilers center Mark Letestu said. "We came out flying in the first period and we were skating and doing the right things, and we capitalized on a couple of good chances."
Shaw scored on the power play at 12:55 to cut the Oilers lead to 3-1. Paul Byron took a point shot, which was stopped by Talbot, but the puck bounced in off Shaw in front.
McLellan challenged for goaltender interference by Shaw on Talbot, but the goal was upheld following video review.
The Canadiens had several scoring chances before the end of the second, but Talbot made four saves to keep the Oilers ahead by two goals entering the third.
"We weren't good enough in the first half of the game," Montreal coach Claude Julien said. "I thought we picked it up in the second half. Obviously, that goal gave us a little bit of life. But we didn't seem to find our legs or even our momentum in the first half, and we seemed to be chasing the game more than anything else."