The Coyotes jumped out to a two-goal lead after 20:00 play, courtesy of Chris Summers and Rob Klinkhammer.
The were able to storm back in the middle stanza, scoring two power play goals. The first came off of Curtis Glencross' stick and Mikael Backlund notched the equalizer at the 15:37 mark.
The Flames took six straight penalties in the third period and the Coyotes were able to take the lead with the man advantage. Shane Doan tipped Keith Yandle's point shot past Joni Ortio at 8:49 with Tyler Wotherspoon in the sin bin.
Mike Cammalleri has six points in his last four games. He posted two assists on Saturday.
The Flames were credited with 37 hits in the contest. Ladislav Smid led the way with nine while Lance Bouma threw eight checks.
The Flames put 32 shots on Mike Smith through 60:00.