Weber made it 2-0 24 seconds into the second. He scored with a one-timer from the left point after Raanta lost a skate blade in the crease and couldn't recover.
"Start of the first period, (the Canadiens score) seconds in; start of the second period, (they score) seconds in," Coyotes coach Rich Tocchet said. "They're killers. You can't give freebie (goals) up to Carey Price. He's too good of a goalie."
Nick Cousins, who played the past two seasons for Arizona, scored 1:17 into the third period to put Montreal ahead 3-0.
"Obviously, it's fun to score against your former team and especially a big goal there in the third to kind of pad our lead," Cousins said.
"He definitely wanted one," said Nick Suzuki, who set up Cousins in the slot with a one-touch pass.