The four-point performance tied a career best for Iginla, who also had four points against Phoenix on Feb. 23, 2003.
The Flames took the lead in the opening period. Iginla knocked in a rebound of a Dion Phaneuf shot from the left point at 18:21.
The Flames increased their lead to 2-0 early in the middle period when Alex Tanguay fired into an open net with Phoenix goal Mikael Tellqvist out of position.
Phoenix answered with two goals within 3 1/2 minutes.
Defender Keith Ballard carried the puck in behind the Calgary net and slid a pass out in front. A trailing Ladislav Nagy shot the puck past Miikka Kiprusoff at 4:28. Mike Comrie lifted in a rebound at 8:01 to tie the game.
The Flames regained the lead when Daymond Langkow scored on a power-play goal, grabbing a rebound from Richie Regehr's shot from the right point at 10:06. On the play, Tellqvist lost his stick and was caught in a tangle of players in front of the Coyotes net.
Iginla added his second tally of the game, and 20th of the season, with a two man advantage when he fired in a 20-foot wrist shot from the slot at 7:12 of the final period.
The Coyotes drew within one at 4-3 when Travis Roche crashed the slot, knocking in a drive off the stick of Oleg Saprykin at 14:55 for his first NHL goal.
Calgary responded when Bryon Ritchie drove in a rebound at 15:50.
Iginla's second assist came on an empty-net goal from the Flames' Tony Amonte in the final minute.
It was the fourth loss in the last five games for Phoenix.
Notes: Day-to-day with a groin strain, Phoenix goalie Curtis Joseph is two wins behind Jacques Plante for fifth place all time with 435. Just ahead is Terry Sawchuk with 447. ... Attendance was listed as 15,551, and the Coyotes have not sold out a home game this season.