Gaudreau stripped Senators defenceman Erik Karlsson of the puck and worked a quick passing play with Mikael Backlund in alone before burying a shot by Mike Condon with 30.2 seconds remaining in overtime.
"I got a stick on the puck there in the neutral zone," said Gaudreau, who also had an assist in the game. "Backs worked hard to pick up the puck and kind of went on a 2-on-0. It's hard to miss those empty nets.
"It was a great play by him and a great win for us."
It's the fourth overtime-winner of Gaudreau's career, and his first goal in 12 games.
"He's going to the All-Star Game here and he's going to come back firing, I'm sure," Monahan said. "Those two points are big for him. He's going to have some confidence and get going."
Monahan and Micheal Ferland also scored for the Flames (24-24-3), who entered the game with one win in the past seven contests (1-5-1), and trailed at least 4-0 in each of the past four games.
The win moves Calgary moves back to the second wild card spot from the Western Conference heading into the NHL All-Star break.
"Tonight I think we've accomplished a few things," coach Glen Gulutzan said. "I thought we did a pretty good job. We were tight-checking tonight. We gave ourselves a chance. The penalty kill was real good.
"More importantly, some guys that needed a little boost got it. Ferls, Johnny, Mony…they all got big goals for us. Our defence played well and our goalie played well.
"It's something we can head off into the break and build off of."
Tommy Wingels and Chris Wideman scored, and Condon made 23 saves for the Senators.
Ferland ended a run of nine straight games with the Flames allowing the first goal when he scored at 9:39 of the second period. Ferland's initial shot was stopped, but he followed up the rebound and shot through Senators defenceman Marc Methot and by Condon for his sixth of the season to give Calgary a 1-0 lead.
Condon made a blocker save off Matthew Tkachuk just a minute-and-a-half later, and stopped Michael Frolik's drive to the net at 16:03.
He couldn't stop Monahan with 2:52 remaining in the period.
Monahan corralled Dougie Hamilton's rebound at the edge of the crease and pulled the puck between his legs to avoid a pokecheck before lifting his team-leading 16th goal over a downed Condon to push the Flames' lead to 2-0.
"Great play by Johnny to turn up there and find Dougie," Monahan said. "Dougie made a good play. The puck was sitting there in front of the net. It was just a reaction. I'm pretty surprised myself but I'm happy it went in.
"I didn't really think about it. It just kind of happened. I think that was a big goal there, a turning point in the game. We needed to maintain that lead and we didn't, but we needed to find a way to win and we did that."
Wingels cut into Calgary's lead just 2:01 into the third when he deflected Fredrik Claesson's shot from the point behind Elliott. The game was Wingels' first with the Senators after being acquired from the San Jose Sharks on Tuesday.
Wideman scored again on a one-timer with 62 seconds remaining in regulation to force overtime.
"Those kinds of things sting but as a group we've been talking a lot lately," Monahan said. "We found a way to bounce back and in OT we showed it."
NOTES: Lance Bouma paced the Flames with five hits … Deryk Engelland finished with a team-high four blocked shots … Captain Mark Giordano led Calgary in ice time at 25:16. Forward Freddie Hamilton and defenceman Jyrki Jokipakka were scratched for the Flames … Forward Garnet Hathaway (upper body) remains injured.
UP NEXT: The Flames will return to action following the NHL All-Star Break on Wednesday, Feb. 1 when they host the Minnesota Wild (8 p.m. MT, SN360, SNW, SN960).
