Garland gave the Coyotes a 1-0 lead at 8:00 of the first period on a shot from along the goal line that deflected off defenseman Nicolas Hague's skate and past Malcolm Subban, who made 12 saves in his first start of the season before leaving after the period with an undisclosed injury.
"We defend well," coach Rick Tocchet said of the Coyotes, who have allowed four goals in three games. "When we start throwing pucks at the net and they go in, we're a good club. Nobody was panicking after two games, we liked the way we played. And to beat Vegas, that's a tough team to play."
Hayton made a one-handed pass to Christian Dvorak to set up Nick Schmaltz's goal for a 2-0 lead with 39 seconds left in the first.
"It was amazing, just getting a feel for the game was great, getting a feel for the speed," Hayton said of his first NHL game. "Overall, I thought it was a good game. It was great to get a little chemistry going."
Theodore cut it to 2-1 with a shot from the top of the right face-off circle with five seconds remaining in the first. Tocchet challenged for goalie interference, but the goal was upheld after video review, resulting in a delay of game penalty against Arizona.
Garland came out of the penalty box after serving the two minutes for the unsuccessful challenge and scored on a breakaway to make it 3-1 at 2:03 of the second.
Niklas Hjalmarsson scored with a slap shot from the right point for a 4-1 lead at 18:28 of the second. It was the defenseman's first goal in 94 games dating to March 5, 2018.
"It's fun to see that go in and feel that you can do something offensively for the team," Hjalmarsson said. "It's not fun after the season to go 82 games and there's a big zero there with the goals. So it feels pretty good to get that out of the way, and hopefully I can maybe squeeze in a couple more."
Marc-Andre Fleury made 13 saves on 15 shots in relief for Vegas.
"It's just a bad game. No reasons and no excuses. We're a good team, and we'll never find excuses, only a solution from here," Golden Knights forward Jonathan Marchessault said.