Brodie's goal with 2:34 left in overtime gave the Flames their third consecutive win and second victory over Chicago in three games this season.
Mikael Backlund scored twice in regulation and Matt Stajan added a goal and an assist for Calgary. Karri Ramo stopped all nine shots he faced in the third period in relief of Reto Berra, who was touched for four goals on 24 shots.
Corey Crawford allowed one goal on 14 shots after he replaced Antti Raanta midway through the second period. Chicago slipped to 0-2-2 in their last four games.