"I felt good," Fleury said. "I felt pretty good. Sometimes mentally, it's a 60-minute game and you haven't played in a long time. I didn't have many shots, so that made it a little easier.
"I thought our team looked good right from the start. Maybe at the start they brought it to us a bit, but we didn't take too long. Our guys found their legs and played very solid. They have a lot of speed on that team, but I thought we matched it, so no problem."
Vegas next plays in the Western Conference round-robin portion of the Stanley Cup Qualifiers against the Dallas Stars on Aug. 3 (6:30 p.m. ET; NHLN, NHL.TV, SN1, ATTSN-RM, FS-SW).
Coach Peter DeBoer has said Fleury and Robin Lehner will get a chance to win the No. 1 goalie job for the Stanley Cup Playoffs during the round-robin.
Reilly Smith scored two special-teams goals in the first period.
While killing a penalty for too many men on the ice, Smith skated down the right wing, received a cross-ice pass from William Karlsson, and scored short side to put Vegas ahead 1-0 at 8:42.
Smith scored on the power play at 16:23 with a shot from the left face-off circle over goalie Darcy Kuemper's left shoulder to make it 2-0.
"I think we were a little sloppy at the start," Smith said. "It seemed like there were a lot of bodies running into each other, especially at the start of the game. I think we did a good job cleaning it up after that."
Video: VGK@ARI: Smith converts on power play for second goal
The line of Nick Cousins, Alex Tuch and Nicolas Roy combined for seven points (two goals, five assists) and a plus-6 rating.
"I thought they were great," DeBoer said. "I thought [Tuch] was all over the ice, and [Cousins and Roy] had really good games too. I think, for sure, our depth was a difference maker tonight, and it's got to be going forward."
Vegas also will play the St. Louis Blues (Aug. 6) and Colorado Avalanche (Aug. 8) in Edmonton, the Western Conference hub city, to determine seeding for the playoffs.
"I'm sure there are some areas we can clean up," DeBoer said. "Overall, considering the circumstances, I thought we came in and did a real good job carrying over a real good camp into the first exhibition game, so we want to build on that."
Kuemper made 17 saves through the second period. Antti Raanta played the third period and made six saves for the Coyotes, who will play Game 1 of their best-of-5 qualifier series against the Nashville Predators at Rogers Place on Sunday (2 p.m. ET; USA, NHL.TV, SN360, FS-TN, FS-A). Arizona is the No. 11 seed, and Nashville is the No. 6 seed.
Taylor Hall scored a power-play goal at 9:14 of the third to get the Coyotes within 2-1, but Tuch scored with a shot from along the goal line at 13:47 to make it 3-1. Cousins scored into an empty net with 13 seconds left for the 4-1 final.
"To me, (the message is) just relax," Arizona coach Rick Tocchet said. "I felt a few guys were a little nervous out there, and that's inexperience. Vegas is a good team to play before the [Qualifier] game, they're a tough team to play against. It was a good first game for us; I'm glad we played them. I thought our third period was better, but we were taking penalties, Vegas was quick and they were winning the 50-50 battles. To me, it's details, discipline."
Video: VGK@ARI: Hall nets PPG amid scramble in front
Hall said he expects the Coyotes to play much better Sunday.
"Yeah, win or lose, we're still eyeing Game 1 as peaking, as that being our best performance," Hall said. "Tonight wasn't the result we wanted, but sometimes it's better to get a kick in the butt and kind of reinforce what we have to do as a team and how we have to play.
"Nashville's going to play similar, they're going to be on us, they're going to be in our faces in the neutral zone. We didn't play the way we wanted to in those areas."
Coyotes forward Nick Schmaltz left the game at 13:14 of the second period after taking a hit from Ryan Reaves.
"We lost Nick, we'll evaluate tomorrow," Tocchet said.
Max Pacioretty, Vegas' leading scorer with 32 goals and 66 points, didn't travel to Edmonton after missing the final four days of training camp. DeBoer said before the game that the forward was "improving quickly."
As part of the NHL Return to Play Plan, a team is not permitted to disclose player injury or illness information.
Video: Smith, Cousins lead Golden Knights past Coyotes, 4-1