"Guys played well, and more importantly than how we play, I thought we had really good energy," said Keefe, who coached the Toronto Marlies of the American Hockey League for five seasons, winning the Calder Cup in 2018. "I was really impressed with my first time being part of the group, the interaction of the players on the bench, the energy, the vibe."
[WATCH: All Maple Leafs vs. Coyotes highlights]
Frederik Andersen made 30 saves, and Pierre Engvall scored his first NHL goal for the Maple Leafs (10-10-4), who ended a six-game winless streak (0-5-1) one day after Mike Babcock was fired as coach in his fifth season with them.
Tyson Barrie and Auston Matthews scored for Toronto, and Ilya Mikheyev had two assists.
"It's a pretty special night for a lot of guys here and a big two points, with everything that's been going on last couple weeks here, and the last 36 hours," Matthews said.
Vinnie Hinostroza scored late in the third period, and Darcy Kuemper made 29 saves for the Coyotes (13-8-2), who had won two straight games by shutout and four of their previous five games.
"The energy level wasn't there," Arizona coach Rick Tocchet said. "I told the boys afterward it was a dud. We haven't had too many this year, but it was a dud. They looked good, but we were out to lunch."
Barrie, a defenseman who scored 14 goals each of the past two seasons for the Colorado Avalanche, gave the Maple Leafs a 1-0 lead at 19:14 of the first period with a wrist shot from the right face-off circle for his first goal with Toronto in his 24th game.
Video: TOR@ARI: Barrie nets first goal with Maple Leafs
Toronto's defensemen were more active in the offensive zone in Keefe's first game.
"I think our whole team was a little more aggressive and had a little bit more speed to the neutral zone, and we had a lot of fun out there," Barrie said. "We kind of felt excited. It was like a fresh start."
The Maple Leafs allowed the first goal in their previous seven games and hadn't led in regulation in eight games, or since a 3-1 win against the Los Angeles Kings on Nov. 5.
"We asked a lot of them, a lot of new things coming at them," Keefe said. "We tried to keep things as simple as possible, but it's still new things coming at them. We came up with a very good effort right to the end."
Engvall, playing in his second NHL game, outraced two Coyotes to the puck in the neutral zone and scored the Maple Leafs' third shorthanded goal of the season to make it 2-0 at 16:49 of the second.
"The feeling to score was unbelievable. I'm just so happy for it," Engvall said.
Video: TOR@ARI: Engvall nets first NHL goal
Matthews, who is from Scottsdale, Ariz., scored his first goal in 10 road games 48 seconds into the third to make it 3-0.
Hinostroza scored with 17 seconds remaining for the 3-1 final.
The Coyotes were 14-3-1 in their previous 18 games against the Maple Leafs.
"I don't think it's a secret, that wasn't our best night as a group," Kuemper said. "We knew there was a big shakeup over there and they were going to come out hungry, and that's what they did. They kept things simple, worked hard, and got rewarded."
They said it
"I was a little behind on a couple of my line changes, so I could speed that up a little bit, but the players took care of me there, and I appreciated that. All in all, it just felt like a hockey game. I didn't know how it was going to be, but the whole day felt comfortable." -- Maple Leafs coach Sheldon Keefe on his debut
"I think we're just excited to get one in the win column. That took a little pressure off ourselves, and we can just continue to build on this. (With) a new coach, we need to work on some stuff that's going to change systematically, but I thought, with everything that he threw at us in a short period of time, we executed that for the most part." -- Maple Leafs forward Auston Matthews
"We got outworked. I don't think anyone was really sharp. This is one we kind of have to flush away. It was just all around not a good game for anyone." -- Coyotes forward Brad Richardson
Need to know
Matthews has four goals and eight assists in the past nine games. ... For the ninth time in their history, the Coyotes were not called for a penalty. ... Forward Barrett Hayton, the No. 5 pick in the 2018 NHL Draft, played his 10th game with Arizona this season, starting the first season of his three-year, entry-level contract.
Maple Leafs: At the Colorado Avalanche on Saturday (7 p.m. ET; CBC, SN1, SNO, SNP, ALT, NHL.TV)
Coyotes: At the Los Angeles Kings on Saturday (4 p.m. ET; FS-W, FS-A PLUS, NHL.TV)
Video: Maple Leafs top Coyotes in Keefe's debut