Hayton made it 2-0 at 11:41 of the third period with a power-play goal, putting in his own rebound after following up Keller's shot from the edge of the right circle.
McBain extended it to 3-0 at 12:30 when Saros went wide to deflect Lawson Crouse's shot from the left circle, leaving the net open for McBain to put in the rebound.
Glass scored with two minutes remaining to cut it to 3-1 before Nick Schmaltz scored an empty-net goal with 37 seconds left for the 4-1 final. It was Schmaltz's third straight game with a goal.
The Predators couldn't score on four power plays in the first two periods and were 0-for-6 overall after going 7-for-17 in their previous seven games.
"[We were] getting all of those goals with really good shots, creating havoc and getting the puck back and making plays off that," Glass said. "We didn't really do that today."
Prosvetov's best save came at 2:30 of the second after Schmaltz's turnover created a Ryan McDonagh breakaway, but Prosvetov made the save and Kiefer Sherwood's follow-up also missed.
"You see all the big goalies in the NHL make those saves, and they come from practice, you have to battle on every shot," Prosvetov said. "Then hopefully you make one in a game, maybe one in 10 games."
Prosvetov's initial concern was getting to Mullett Arena. The normally 90-minute drive from Tucson took so long, he feared he might arrive too late to play.
"I think the fact Ivan was called up today brought a little bit of urgency into our room, [because] we rely on the other two (Karel Vejmelka and Connor Ingram) so much," Arizona coach Andre Tourigny said. "You saw it in the first 50 minutes, we were way more stingy defensively and we did a lot of good things."
NOTES: Arizona allowed four power-play goals in its previous three games. … Prosvetov was 0-3-1 with a 4.19 goals-against average in four NHL starts across the 2020-21 and 2021-22 seasons. … Schmaltz has seven goals in his past nine games. … Keller has 65 points (26 goals, 39 assists) to tie his career high set in 2017-18. … The Coyotes are 16-11-3 at home.