It was only the third win in the last nine games for Calgary (15-12-2), which had to grind it out after losing in a shootout the night before in Toronto.
"Any win’s good," said Monahan, who scored his 16th and 17th goals of the season. "Whatever way you get it, you take the two points and move forward.
"We’ll be happy to have Chucky (Tkachuk) back next game and we’ll build on it from there."
Tkachuck served a one-game suspension for spearing Toronto’s Matt Martin. It was a second suspension this season for the skilled but sometimes undisciplined 19-year-old.
Rittich made only his second career start and improved to 2-0-0 after beating Colorado in his season debut on Nov. 25.
The Czech goaltender made 35 saves, outduelling Carey Price who stopped 34.
"We did a lot of good things but the team over there was resilient," said Price. "Their goaltender played well for them and they found a way to win."
The Flames trailed 2-1 going into the third period but tied it at 7:49 on a goal from fourth-line winger Garnet Hathaway that needed two video reviews. The officials first ruled goaltender interference when the puck went in with Hathaway and two linemates jamming at the puck in the crease.
Flames coach Glen Gulutzan challenged the ruling, which was then overturned.
"Janko (Mark Jankowski), Benny (Sam Bennett) and I were buzzing," said Hathaway, who got his first goal of the season and his second in 46 career NHL games. "Benny made a nice play behind the net to bring it out in front and we were all jamming at it.
"That was a big goal for our line."
Monahan finished a 2-on-1 break by taking a feed from Johnny Gaudreau and beating Price from the slot in overtime.
Daniel Carr and Phillip Danault scored for Montreal (13-13-4), which is 5-1-1 in its last seven games.
"It wasn’t a bad game, but little details cost us the game," said Montreal coach Claude Julien. "We need to win more battles along the boards and in front of the net.
"That’s an area we need to improve quickly."
Each team scored from the side of the net in the first frame. Monahan’s first shot hit the side, but he picked up the rebound, spun around and sneaked the puck inside the near post at 6:31. Then Carr backhanded a shot in off Rittich’s back from close range at 8:36.
It was the sixth point in four games for Carr since he was called up from AHL Laval last week.
Max Pacioretty broke down the right side and backhanded a pass in front that Danault tipped over Rittich’s glove at 10:10 of the second period.
The Flames had a chance for the lead with a two-man advantage for a full minute starting at the 12:47 mark but couldn’t score.
With Tkachuk’s one-game suspension, Freddie Hamilton moved into the Calgary lineup. Curtis Lazar played his 200th game. Jaromir Jagr sat out a second straight game with a lower-body injury.
Montreal started Jakub Jerabek on defence, bumping Victor Mete. Jonathan Drouin skated Thursday morning but still missed a fourth game, the last two with an illness.