Jordan Martinook had a chance to tie it at 10:35 of the second period, but Vejmelka made a left-pad save before also stopping the forward's rebound attempt.
Kochetkov then made a highlight-reel save to keep it a one-goal game at 18:54, reaching back with his stick along the goal line to deny a one-timer from Travis Boyd off a cross-crease pass from Gostisbehere.
However, Crouse extended the lead to 2-0 at 1:52 of the third period when J.J. Moser's point shot hit Guenther in front and caromed to Crouse for a backhand.
Nick Bjugstad scored on a rebound in front at 9:14 to make it 3-0, and Crouse shot into an empty net at 17:53 for the 4-0 final.
"We started playing more to our identity in the second period," Crouse said. "They're always sending it around, getting it to the [defense], quick shots. We talked about what we had to clean up and went out there and did our best. Obviously, in the third, it paid off for us."
NOTES: Tourigny coached his 100th NHL game (32-59-9). … Coyotes defenseman Jakob Chychrun, who made his season debut in a 4-3 shootout loss to the Nashville Predators on Monday, had the secondary assist on Crouse's empty-net goal. ... Schmaltz's goal was his first this season. He returned Monday after missing 15 games with an upper-body injury. ... Carolina was 0-for-4 on the power play and is 0-for-20 in its past six games. ... Kochetkov, who lost in regulation for the first time in eight NHL games (seven starts), signed a four-year, $8 million contract earlier Wednesday. It has an average annual value of $2 million.