Having not scored since opening night, Byron gave the Flames a 1-0 lead 11:06 into the first period then scored early in the second to Calgary the lead again at 2-1.
Trailing 3-1, Arizona stormed back to tie it only for the Flames to take the lead for a third time at 16:23 of the second period on Mark Giordano's sixth goal.
The Flames captain was set up neatly by TJ Brodie, who faked a shot at the point, moved in and after freezing goaltender Mike Smith, slid the puck down low to Giordano, who wired a one-timer just under the crossbar to make it 4-3.
Nursing that one-goal lead, Calgary got some insurance at 12:23 of the third when David Jones' flung the puck toward the front of the net and it banked in off of the leg of Connor Murphy. It was the third of the season and third in as many games for Jones.
Sean Monahan also scored for Calgary (10-6-2) while Sven Baertschi added two assists, his first points of the season. The Flames, who have won five of their last seven, were kicking off a five-game homestand that will continue Saturday night against the Ottawa Senators.
Shane Doan, Antoine Vermette and Joe Vitale scored for Arizona (6-9-1), which has lost three straight. The Coyotes will continue their three-game road swing through Western Canada with a game Friday night in Vancouver.
Arizona drew within one at 3-2 when Vermette converted Doan's cross-ice pass for a power-play goal at 9:58 of the second period.
Less than four minutes later, the Coyotes tied it on a goal that Jonas Hiller was visibly frustrated with.
Vitale's wrist shot from the corner of the rink that snuck inside the goal post on Hiller, stunning the announced crowd of 18,698.
Starting for the seventh time in the last eight games, Hiller finished the night with 20 stops to improve to 8-4-1.
Smith, who has gotten off to a shaky start this season, had 20 saves to see his record fall to 4-9-0.
Calgary opened the scoring at 11:06 when Byron snapped a shot past Smith from the slot.
Getting the puck back to Byron with some hard work was Baertschi, who picked up his first point of the year.
Playing his fifth game since being recalled from Adirondack, Baertschi set up Byron's second goal, gaining the blue-line on a rush and then sending a pass across the slot that Byron cradled then wristed in to the net.
Notes: Calgary improves to 5-0-1 in games after a regulation loss... Not dressed for Arizona was C Brandon McMillan, who sat out his first game of the year, D Chris Summers and D Brandon Gormley... Healthy scratches for Calgary was LW Brandon Bollig and RW Devin Setoguchi. Including last year's playoffs with Chicago, Bollig hasn't scored in his last 50 games. Setoguchi has no points in 10 games this season. Both players were acquired this past summer... The Flames improve to 7-1-2 when scoring first... It was the first of five meetings between the two teams this season.