The Maple Leafs thought they tied the game with 39 seconds left, but the goal was disallowed after the situation room determined there was a missed game-stoppage because of a hand pass. Lawson Crouse then made it 4-2 at 19:49 when he shot into an empty net.
Karel Vejmelka made 26 saves for the Coyotes (1-2-0), who had allowed 12 goals in their first two games.
"Pressure, I think that's the big thing," Coyotes forward Christian Fischer said. "In the third period, you give [Auston] Matthews, [Mitchell] Marner, [William] Nylander, I could go on and on, you give those guys space, they're going to kill you. They're going to make plays and score goals. Then you start chasing them and drag them down [and they] get some power plays. You don't want to go against that power play too many times. A lot of lessons to be learned. I think the best thing we did was put pressure on them, and that's exactly what we didn't do in the third and it almost cost us."
Nylander and Marner scored, and Erik Kallgren made 15 saves for the Maple Leafs (2-2-0) in his season debut after being recalled from Toronto of the American Hockey League on Saturday.
"Inconsistent," Maple Leafs defenseman Morgan Rielly said. "Early on there was times where we had the puck a lot, and about halfway through the game we kind of went into a lull and we didn't have much going on and they controlled the play for the most part, and then we seemed to find our game again. If you look at the first period, we had the puck a lot, (but) we didn't generate a ton from the interior."
Nick Ritchie gave the Coyotes a 1-0 lead at 19:34 of the first period when he redirected a pass in the slot from Gostisbehere on the power play.
Fischer made it 2-0 at 8:32. Nick Bjugstad won a battle for a loose puck against Rielly behind the net and passed to Fischer, who was at the top of the crease.
Nylander cut the lead to 2-1 on the power play at 12:54 of the third period. John Tavares' shot from the high slot hit Josh Brown, and Nylander shot past Vejmelka's glove from below the left hash marks.
Marner tied it 2-2 at 13:18. Troy Stecher, who was behind the net, attempted to pass to J.J. Moser in the slot, but he mishandled the puck and Marner scored with a backhand in the slot.
"We competed hard, and we scored a big [go-ahead] goal," Arizona coach Andre Tourigny said. "When you're under pressure like that and you lose the momentum, it can be tough mentally, and our guys were resilient. Really happy about the way it happened on the bench. We had leadership, guys who stood up and talked, so that was good."