Iginla nearly duplicated the singular feat of Mario Lemieux by scoring on the power play, at even strength, shorthanded and into an empty net. All that was missing was a penalty shot.
Last season's Art Ross and Maurice Richard Trophy winner, Iginla got things going with an unassisted goal at 6:43 of the first period, beating goaltender Sean Burke with a wrist shot from the slot.
Phoenix defenseman Deron Quint tied it 71 seconds into the second, also on the power play, as he flicked a rebound past goalie Roman Turek's right side.
Just 76 seconds later, Iginla scored from outside the crease to put Calgary ahead for good. He struck again while the Flames were killing a cross-checking penalty on Blake Sloan.
After rookie Ramzi Abid could not control the puck in his own zone, Iginla moved in and put a low shot past Burke for his third shorthanded goal of the season.
Ladislav Nagy answered on the power play midway through the third period when he pounced on a rebound of defenseman Teppo Numminen's shot off the post and beat Turek for his 14th goal.
But Iginla secured the win with a minute to play, getting a feed from Turek before finding the open net.
Iginla spoiled the return of Burke, who made 29 saves after missing 19 games with a sprained knee. The loss ended the Coyotes' five-game unbeaten streak snapped.