Corey Perry scored, and Jake Allen made 27 saves for Montreal (12-7-7), which was coming off a 5-1 at the Vancouver Canucks on Wednesday and had its six-game point streak end (3-0-3).
"Points are crucial," Allen said. "They came out and played a heavy, hard game. We knew they were going to do that with the new coaching change. We just couldn't find our groove tonight. I think we'll get back to the drawing board tomorrow. I think we've played some really good hockey as of late and we'll get to it again Saturday."
The teams will complete a two-game set here Saturday.
"You want to win hockey games, (and) in this division you need four good lines, and three good (defense) pairs, and your goaltender to make some big saves, or good saves," Sutter said. "I think that we're going to have to work at that because we'll play a much better team on Saturday night than we played tonight."
Leivo finished a passing play with Ryan and Bennett on a rush 1:52 into the second period to put the Flames ahead 1-0.
Leivo extended the lead to 2-0 at 13:55, shooting by Allen's blocker following a turnover up the middle in Montreal's zone by defenseman Shea Weber. Leivo has three goals in his past three games after not scoring in his first 18 games.
"It feels great," said Leivo, who also hit the post on a setup by Ryan at 10:36 of the second. "I don't think I get those goals without those linemates and the way we played. They worked hard, created some turnovers, and I was just standing in the right spot. When you're ready for it, sometimes the puck goes in. I've got to give credit to my linemates."
Bennett is seeking a fresh start under Sutter after reportedly requesting a trade in late January.
Perry tipped Weber's point shot and beat Markstrom five-hole to cut the lead to 2-1 at 10:15 of the third period.
"Sometimes it's just not there," Canadiens forward Jonathan Drouin said. "When it's not there as a team, we have to find those ways to do the little things right and make sure we're on the same page. It's not going to be a pretty night. We're going to get nights like that when it's not flowing the way it should be, but we'll get better at that where we've got to figure out our game when we're not having it."
NOTES:Montreal defenseman Ben Chiarot did not play after fracturing his right hand in a fight against Vancouver forward J.T. Miller on Wednesday. ... Canadiens forward Tomas Tatar played his 600th NHL game.
