Calgary answered back in the opening minute of the second period, as Elias Lindholm put a Matthew Tkachuk feed home at the 14-second mark to pull the Flames even, 1-1.
The visitors took a 2-1 lead late in the middle frame as found Tkachuk found Lindholm again for his second of the night, putting a rolling-puck snap shot past Crawford at 16:01.
The Blackhawks pushed late with an empty net, but fell to the Flames 2-1.