Captain Shane Doan scored the game winning goal for Phoenix at 8:49 on the power play in the third period from defenseman Keith Yandle and veteran faceoff specialist Antoine Vermette.
The Coyotes got two first period goals from Chris Summers and Rob Klinkhammer, while the Flames responded in the second period on goals from Curtis Glencross to Mikael Backlund. Glencross and Backlund both scored on the power play.
Summers, a defenseman and native of Ann Arbor, Michigan, scored his first National Hockey League goal in his 34th career game. Summers was the Coyotes first round pick, 29th overall in the 2006 National Hockey League entry draft.
Glencross meanwhile scored his first goal in his third game back since missing 29 games with an ankle injury. Both of the Flames goals were assisted by Michael Cammalleri.
The Flames drop to 27-34-7. They are 14 points back of the Dallas Stars for a playoff spot in the Western Conference and next play host to the Buffalo Sabres on Tuesday.