MacKenzie Weegar put the Flames back ahead 2-1 at 17:50, scoring on a wrist shot from the right point through traffic.
Iafallo tied it 2-2 at 11:19 of the second period when he tipped a Dylan DeMelo point shot by Markstrom's blocker.
Iafallo was acquired alongside forwards Gabriel Vilardi, Rasmus Kupari and a second-round pick in the 2024 NHL Draft in a trade with the Los Angeles Kings for forward Pierre-Luc Dubois on June 27.
Rasmus Andersson gave Calgary a 3-2 lead at 13:49 with a short-handed goal after he shot by Hellebuyck's blocker from the high slot on an assist from Lindholm.
Andersson's goal came 12 seconds after Markstrom stopped Scheifele's one-timer from the slot.
"‘Marky’ was awesome," Mangiapane said. "He's a big part of the team. When you see him dialed in like that, it's awesome to see. I think shots were probably 12-2 to start the game or something like that and he was standing on his head to keep us in it throughout the whole game. It was a big game for him and we're going to need a lot more of those from him."
Scheifele tied it 3-3 at 14:18 of the third with a shot short side over Markstrom's glove.
"I thought we played a really solid game," Scheifele said. "Markstrom made some pretty unbelievable saves. He held them in there. But a lot of good things. I think we've got to be positive here. It's the first game, and first game really with our entire lineup. We played a solid game. We can't dwell on the result too much, but keep building."
NOTES: Connor tied the NHL's longest goal streak in season-opening games, joining Cam Atkinson of the Columbus Blue Jackets (2014-15 to 2019-20), Yvan Cournoyer of the Montreal Canadiens (1973-74 to 1978-79) and Mud Bruneteau of the Detroit Red Wings (1940-41 to 1945-46). ... Vilardi had seven shots in 21:04 of ice time in his Jets debut. ... Scheifele has a franchise-best nine goals in season openers (12 games). ... Mangiapane is three goals away from 100 in the NHL. ... Scheifele and Hellebuyck each signed a seven-year, $59.5 million contract on Monday (average annual value of $8.5 million).