Dougie Hamilton had a goal and an assist for the visitors (35-29-10), who led 2-1 late in the second period before the Coyotes rallied for the win at home to snap a two-game losing streak.
"We have to be a lot smarter," said Calgary captain Mark Giordano. "We gave them their second and third goals. We just didn't manage the puck when we had to in key moments in that game. That was the difference really.
"We were putting pucks to the net, but puck management at critical times … it bit us tonight. We've got to try and get that one at least to overtime."
Sean Monahan also scored for the Flames, who are six points behind the Anaheim Ducks (37-24-12) for the second wild-card playoff spot in the NHL's Western Conference.
The Flames return home from their quick two-game road trip to face those very Ducks on Wednesday at the Scotiabank Saddledome.
"Every game that we lose, there's less and less room for error, if any now," Giordano said. "We've got to go into every game basically knowing that we have to get to get the two points."
Mike Smith made his return to the Gila River Arena in Glendale, Ariz. He played the past six seasons for the Coyotes before being acquired by the Flames last June.
"I think as you get older, you start to cherish the times that you have a chance to play on good teams and play on a playoff team," Smith said. "That was my feeling coming into this season. On paper, you look at our team and we should be a lot further ahead than we are. I think that's the frustrating part about this season and how it's gone is we've underachieved by a lot."
Richard Panik also scored for the Coyotes (24-37-11), while goalie Antti Raanta stopped 42 shots.
Like they did one day earlier in Las Vegas in a 4-0 loss to the Golden Knights, the Flames came out strong in the first period but weren't able to score.
Calgary outshot Arizona 16-9 in the opening frame on Monday, but Raanta was solid in net to keep the Flames off the scoreboard.
Hamilton rang a shot off the post during an early powerplay for the visitors, while Raanta made a great glove grab to deny a scoring attempt by Troy Brouwer on another man advantage.
Not to be outdone, Smith made a pad save to stop a breakaway attempt by Jordan Martinook before sprawling out on the ice and stacking his pads to deny a point-blank chance by Derek Stepan.
The Coyotes opened the scoring at 5:15 of the second stanza when Panik one-timed a cross-ice pass from rookie forward Clayton Keller past Smith.
Hamilton snapped a shot to the short side past Raanta at 9:05 to pull the Flames into a 1-1 tie. Hamilton now leads all NHL defencemen in goals with 16.
Monahan then picked up a rebound in the slot and deposited it into the net at 15:28. Raanta was out of position after sliding across his net to stop the initial shot by Johnny Gaudreau.
Domi pulled the homeside back into a 2-2 tie with 2:29 to play in the second when he intercepted an errant pass by Micheal Ferland in the Calgary zone before quickly firing a wrist shot to the blocker side past Smith.
"At the end of the second period there, we turned one over just inside the blue-line and it cost us," said coach Glen Gulutzan.
Although Calgary outshot Arizona 12-7 in the final frame, it was Ekman-Larsson who notched the game winner when his shot deflected off defenceman Travis Hamonic's stick and into the top corner behind Smith.
"We're trying to get the puck deep and we don't execute and they come back and score," Gulutzan said. "That's all it took."
