Zach Hyman had two goals in regulation for Toronto, and Alexander Kerfoot added two assists. Jack Campbell stopped 35 shots.
Campbell, picked up in a trade with Los Angeles last week, is 2-0-1 in three starts, all overtime games.
''It's been incredible,'' Campbell said of the first week with his new club. ''I can't speak enough about the character of the room. The guys are so welcoming.
''I was pretty nervous going into such a solid team.''
Carl Soderberg and Christian Dvorak scored for Arizona. Adin Hill made 38 saves.
''It (stinks) , we think we won there for a minute, but that's the way it goes,'' Hill said. ''It was a hard-fought game.''
Kapanen, who was forced to sit out a game for being late to practice earlier this month, beat Hill high on a partial breakaway for his 11th goal of the season. The overtime winner ended a 13-game goal-less drought for Kapanen.
''I think everyone here knows that's my go-to move,'' Kapanen said. ''That's been working for me and I decided to try it and (hope) it'd go in. It did.''
Hyman took an offensive-zone tripping penalty with 34.3 seconds left in regulation, and it looked as if Jakob Chychrun had won it after the Arizona penalty expired in overtime.
After the Coyotes celebrated on the ice, the goal was overturned after video review found that Derek Stepan interfered with Campbell.
''I usually don't get too emotional in the game,'' said Campbell, who didn't leave his crease after the puck went in. ''The guy bumped me pretty good.''
Arizona pushed ahead 2-1 late in the second. Hyman scored his second goal of the game and 16th overall 1:33 into the third when he took a no-look, one-knee pass from John Tavares in the slot.
''I'm doing the same thing I've been in years past,'' said Hyman, who's just five goals off the career-high he set last season despite missing the first 19 games of 2019-20 with a knee injury. ''I'm just being more patient and more confident with the puck.''
Campbell made a nice stop on Taylor Hall midway through the third period on an Arizona power play before Hill robbed Kerfoot on a short-handed breakaway on the same man advantage.
Campbell was called upon again on a 2-on-1 with seven minutes to go - stopping Soderberg's initial shot and the follow-up with his glove, even though he couldn't find the puck as it bounced in the air.
Toronto got a late power play when former Leafs winger Phil Kessel went off for tripping. But Hill denied Auston Matthews on a hard shot.
With not much happening at either end through most of an uninspired opening 40 minutes, Soderberg scored his 14th goal of the season with 46.8 seconds left in the middle period off Arizona's third effort on a scramble in front of Campbell.
The Leafs goalie, who was making his third straight start with No. 1 netminder Frederik Andersen (neck) still not ready to return, had to be sharp earlier in the period, making glove saves on Lawson Crouse, Chychrun and Hall. Campbell was acquired along with forward Kyle Clifford in a trade last week with the Los Angeles Kings for winger Trevor Moore and two draft picks.
Following Monday's 3-2 victory over the Montreal Canadiens where Antti Raanta made 26 saves, Arizona turned to Hill - the club's third-stringer - with Darcy Kuemper still out injured.
He made a nice stop on another Matthews one-timer on the power play in the second before the center blasted another shot off the post.
The Coyotes jumped in front just 64 seconds into the opening period on the game's first shot when Oliver Ekman-Larsson's blast hit teammate Connor Garland in front. The puck popped into the slot to Dvorak, who buried his 17th with Campbell out of position.
The Leafs scored at 14:23 when Tyson Barrie's shot hit Hyman in front and went in for his 15th.
Campbell had to be solid at the other end late in the period with three nice stops to keep things tied - one off the stick of Hall and two from Niklas Hjalmarsson.
''The boys have played great,'' Campbell said. ''It felt like a playoff game out there. Both teams didn't give each other much, but Kappy made a great play to seal the deal.''
NOTES: Tavares and Leafs defenseman Justin Holl played after sitting out Monday's practice with an illness, while winger William Nylander, who sat out the last two games sick, was back in for Toronto. Hyman was also in the lineup despite skipping Monday's session with what the team called ''bumps and bruises.'' ... Toronto started with a line of Nylander, Matthews and Mitch Marner - an expensive trio that accounts for nearly $29.5 million in annual average value on the salary cap - but they couldn't generate much of anything. ... Tuesday marked the start of a stretch of four games in six nights that will also see Toronto host Dallas on Thursday before visiting Ottawa on Saturday and Buffalo on Sunday. Looking a bit further ahead, the Leafs play seven games in the next 12 days as they continue to battle for a playoff spot in the Eastern Conference.
Coyotes: At Ottawa Senators on Thursday for final game of four-game trip.
Maple Leafs: Host the Dallas Stars on Thursday.
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