Mike Smith stopped 35 of 39 shots in the win, returning to the lineup after missing two games and two periods of another.
Brandon Manning, Sean Couturier, Ivan Provorov, and Nolan Patrick answered for the Flyers.
Brian Elliott, playing his first game against the Flames since leaving the team, stopped 26 of 31 shots.
"When guys step up and do little things, it means a lot to the guys on the bench," Monahan said. "So I think tonight was a big win, Smitty stood on his head again like always, and we're leaving here with two points that we came for."
It was the Flyers who opened the scoring early when just 92 seconds in when Manning's shot from the blue line went up, off of Smith's shoulder, off the post, off of Smith's back, and in.
But Gaudreau, who grew up just outside of Philadelphia, answered in beautiful fashion. Showing off in front of family and friends, he received a pass from Micheal Ferland, and was off to the races, bearing down on Elliott on a breakaway. He made no mistake, shooting it in between Elliott's pad and blocker short-side to tie the game.
The Flyers got the lead back, though, on the powerplay. Jakub Voracek's shot led to a rebound Smith couldn't corral, and Couturier was there to roof the puck right over him.
Shortly after, Provorov gave the Flyers a multi-goal lead with a seeing-eye shot from the top of the circle, over Smith's shoulder and in.
"I felt better as the game went on," Smith said. "The first few goals were just kind of crazy ones that found their way in.
"Fortunately I was able to get back in there and kind of get some momentum as the game went on and felt like I was solid from then on in."
The Flames - and Monahan, specifically - came back with a vengeance in the second period, scoring three powerplay goals.
"He's got a great shot," Gaudreau said of his longtime linemate. "Makes my job pretty easy, I just find him in the slot, and he's got a world class shot there. I always give him a hard time, he hasn't had a hat-trick, he's got like 100-something goals, and he got his first one tonight. So really happy for him, he deserves it, and he played well tonight."
Video: Monahan has power-play hat trick in Flames' OT win
Coach Glen Gulutzan also had plenty of praise for Monahan.
"He's a consistent scorer, right," he said. "He's not surviving on the big games all the time. Good for him. I've coached some good players, he's really got a nose for the net, I haven't seen a young guy like that that just can track pucks and he's got a great release. Just always seems to know where to go when the loose pucks are popping around.
The third period was goal-free, although there were a number of chances for both teams, including Mark Jankowski and Sam Bennett each ringing a shot off the post, and the Flyers having a powerplay to work with in the final two minutes of regulation.
"It's a step today," Gulutzan said. "I mean a kill with two minutes on the clock and you need a big one when your kill has been struggling all season, that goes a long way.
"Our powerplay came through today big time for us, got us back in the game, our penalty-kill got us a big kill at the end when we really needed it. I think that's going to help our confidence moving forward."
Frolik ended it in overtime, snapping the puck past Elliott off of a perfect pass from Mikael Backlund.
"We got a little two-on-one," Frolik said. "And great pass by (Backlund), it was sauced over his stick, and I just tried to shoot it, and it went in.
"We were always talking before the game, we need this, kind of turn this around a little bit, and make sure we play every night the same way. And we were a little bit kind of up and down, and I think this is a great win, and hopefully can get us confidence."
Gaudreau extended his point streak to nine games and goal streak to five, both career highs. He has six goals and 17 points over that time … Monahan's first career hat-trick came in his 10th career multi-goal game … Ferland has a six-game point streak going … The Flames remain undefeated when a game goes past regulation this season.
The Flames will be taking the train to head to Washington, D.C., for a 5 p.m. MT clash against the Capitals on Monday. They'll continue their season-long road trip from there, with stops in Columbus (Wednesday, 5 p.m. MT), Dallas (Friday, 7 p.m. MT) and Colorado (Saturday, 8 p.m. MT).