Sutter, who replaced Geoff Ward on March 4, was 107-73-15 with 15 ties in three seasons as Calgary coach from 2002-06, including advancing to the Stanley Cup Final in 2004, and was their general manager from 2003-10.
"I think [our confidence] is way up," Monahan said. "We're a lot sharper right now. We're playing together. When you're playing faster and moving your feet it's going to generate a lot more looks, and that's what we're getting right now."
Mikael Backlund scored, and Jacob Markstrom made 24 saves for the Flames (13-12-3), who have consecutive victories for the first time since a season-long three-game winning streak Feb. 6-11.
Calgary swept the two-game set with Montreal after winning 2-1 on Thursday.
"You've got to play with a little bit of swagger and a little bit of confidence when you play this game against these teams," Flames forward Andrew Mangiapane said. "We have to go into every game with the mentality that we're going to come away with the two points. It's just the last two games here, but we have to keep building on it and not get too high or not get too low. You've got to keep building on the wins."
Jeff Petry scored, and Carey Price made 33 saves for the Canadiens (12-8-7), who have three wins in the past 12 games (3-4-5).
"We know we can be a good team," Montreal forward Nick Suzuki said. "These points are going to matter a ton coming down the stretch. We're over the halfway mark, so we need to pick up points. We need wins. It's got to come fast for us. We need to start building our game. Just have to take it to the next step."
Monahan put the Flames up 1-0 at 5:38 of the first period when he cut from behind the net and spun before shooting over Price's glove for his 200th NHL goal. Monahan, the No. 6 pick, leads the 2013 NHL Draft class with 201 goals in 567 games. Nathan MacKinnon, the No. 1 pick, is second with 196 goals in 546 games for the Colorado Avalanche.
"It's pretty cool," Monahan said. "It's special when you win and you have a milestone. It's something I'll remember here for a little bit."