Andrew Mangiapane scored his ninth goal for Calgary (7-3-3), which had its five-game road winning streak end. The Flames lost their second straight after going 7-0-3 during a 10-game point streak.
"We're a team that didn't make the (Stanley Cup Playoffs) last year," Flames coach Darryl Sutter said. "We're trying to stay in the race, that's the key. There's not enough difference in the League, I'll tell you that right now, that every game is a tough game.
"We're not a team that's going to score a lot so you're going to have to do a lot of grinding and banging away, and I think we did more of that tonight than we probably did the last couple of games."
Jacob Markstrom made 24 saves for Calgary, which began a seven-game, 11-day road trip.
"It's a long road trip and we wanted to start with a win, and I thought we had the game where we wanted to," Markstrom said. "But obviously we couldn't finish it."
Suzuki made it 3-2 with Montreal's second power-play goal at 12:08, banking a shot in off Markstrom's pad as he began to cut behind the net.
"He's coming down the wing, not really any backside pressure, I felt like I played a little aggressive, outside the post," Markstrom said. "Guys usually slide it five-hole there so I went down and he's either going to go around, wrap it, I've got to rotate. And I lifted my left leg and it hit my right leg while I was rotating. I needed to get inside the post and to the other side, and it went in."
Suzuki had tried to score on a shot from behind the goal line during 4-on-4 play but Markstrom made a glove save to stop play at 10:38.
"He had some words," Suzuki said. "I think he just said it wasn't going to work on him, but then the second one went in, so he gave a little tap after two shifts later."
Jake Evans scored into an empty net at 19:53 for the 4-2 final.
"He's been playing great," Suzuki said of Evans. "You can see when he's confident he's making plays, skating really well, great on the PK. We definitely need him to play great and he's been stepping up in big moments, and good to get rewarded with the goal at the end."
Mikael Backlund gave Calgary a 1-0 lead at 2:02 of the first. He deflected Trevor Lewis' wrist shot from the left point while he was being covered by Canadiens defenseman Chris Wideman in front of the net.