Mikael Backlund scored twice for the Flames (19-27-7), who won three straight games for the first time this season. Matt Stajan had a goal and an assist, Lance Bouma also scored and Lee Stempniak set up three goals.
Neither of the starting goalies played particularly well and both were replaced in net by their respective head coaches.
Antti Raanta drew the start in net for Chicago, but was pulled after he stopped just 11-of-15 shots he faced through 1 1/2 periods of play. Corey Crawford came on in relief and stopped 13 consecutive shots before giving up Brodie's OT goal.
The Flames held a pair of one-goal leads in the first period before the Blackhawks responded both times to tie up the game 2-2 after 20 minutes of play.
Calgary jumped out to a quick start as captain Mark Giordano rang a shot off the post 37 seconds into the game before Backlund scored his 10th goal of the season at 1:13 of the first.
The Blackhawks evened the score at 5:53 when Kane converted a nice backhand pass from Kris Versteeg from behind the Calgary net.
The Flames regained their one-goal lead at 11:34 when Bouma's weak wrist shot hit Raanta in the shoulder before bouncing up and over the Chicago goalie and into the net.
Once again, the Hawks drew even on a goal by Hossa with 3:25 remaining in the opening frame. Patrick Sharp appeared to score the goal, but his stick hit Hossa's, which actually directed the puck into the net behind Berra, who made a great pad save to stop the initial shot taken by Jonathan Toews.
David Jones made a nice play to set up Stajan for his ninth goal of the season at 3:21 of the second before Backlund converted a feed from Stempniak for a short-handed marker at 10:25 to put the Flames up 4-2.
The Hawks pulled back to within one when Smith's long wrist shot bounced off of Berra's catching glove and into the net. Hossa then tied the game at 4-4 when he swatted a rebound past Berrra at 17:33 of the second while Chicago was killing off a penalty.
The Blackhawks outshot the Flames 9-7 in the third period, in which neither Ramo nor Crawford surrendered any goals.
In overtime, Crawford stopped a shot from the slot by Calgary forward Sean Monahan. Brodie then picked up the rebound and scored a wrap-around goal to emerge as the hero.
Notes: The Flames are in the midst of a five-game homestand, which equals their season high. They also played five straight games at home from Dec. 23 to Jan. 3 when they went 1-4. … The Flames will also host the San Jose Sharks on Thursday and the Minnesota Wild on Saturday. … It was Backlund's first multi-goal game since Oct. 21, 2010 when he scored both times for Calgary during a 4-2 loss to the Red Wings in Detroit. … With an assist on Backlund's second goal, Giordano extended his point-scoring streak to seven games, a career high.