Nick Suzuki had a goal and two assists, and Mike Hoffman had a goal and three assists for the Canadiens (14-34-7), who have won six of seven. Andrew Hammond made 30 saves.
"That's a big one, beating a team like that," Chiarot said. "They're at the top of the standings and you can see why. They don't give you any room out there. That's a big one for the confidence of the team, knowing we can play with anybody. That's a big one for the character of the team, I think."
Elias Lindholm and Andrew Mangiapane scored, and Jacob Markstrom made 30 saves for the Flames (32-14-7), who would have set their record with a 12th straight home win.
"I think lately at home we haven't been, even though we're winning games, we haven't worn teams down and stuff like that like we did a couple weeks ago," Lindholm said. "We need to get back to that. That's how we want to play. Wear teams down and play in the O-zone as much as possible. Right now, we're turning pucks over and playing too much up and down."
Chiarot one-timed a cross-ice pass from Rem Pitlick by Markstrom at 1:03 of overtime for the win.
"He had a great game," Suzuki said. "He's our horse back there and he's had a great season so far. For him to get the OT winner, I thought it was well deserving."