Fleury won his NHL-leading 22nd game of the season for the Golden Knights, who were 7-10-1 after their first 18 games. They are 6-1-0 against the Coyotes since joining the NHL at the start of last season.
The win moved Vegas into a tie with the Calgary Flames atop the Pacific Division at 50 points apiece; the Flames have played three fewer games.
"We had a tough start to this season and some ups and downs," Fleury said. "[Now] we're more consistent and we have a chance to win every night and we've been climbing the standings."
Stastny scored in his third straight game at 3:35 of the first period to give Vegas a 1-0 lead. He has five goals in 12 games after missing 29 straight because of a lower-body injury.
After Nick Schmaltz's tying goal at 11:06 of the second was overturned when Vegas coach Gerard Gallant successfully challenged that Vinnie Hinostroza was offside, Smith made it 2-0 at 16:03 from the edge of the crease off a tipped shot by Brad Hunt.
Galchenyuk scored during a 5-on-3 power play to cut it to 2-1 at 16:57 of the second. The Coyotes went 1-for-6 with the man-advantage.
"We were patient," Fleury said. "I think sometimes when things don't go in, teams open up and take more chances. We were patient and stuck with our game plan and didn't give them too much, and it paid off."
Eakin scored unassisted at 4:21 of the third to give the Golden Knights a 3-1 lead.
"That third goal's a huge goal for us, obviously, for our confidence. I liked the way we played all night," Gallant said.
Pirri's fifth goal in six games since being called up from Chicago of the American Hockey League made it 4-1 at 6:58.
"Things are going good. I've always believed in myself and I'm getting the opportunity," Pirri said. "When you get the opportunity, you've got to reward the people who put you in the spot."
Carpenter scored shorthanded at 11:44 for the 5-1 final.
