The Coyotes won for the first time this season and handed Nashville its first setback in its home opener.
Schlemko, Ray Whitney and Lauri Korpikoski produced goals in a span of 6:45 to get Phoenix started and Mikkel Boedker made it 4-0 midway through the second.
Jason LaBarbera, getting his first start of the season, made 23 saves in the victory.
Pekka Rinne surrendered four goals on 30 shots.
The win by Phoenix left Columbus, Winnipeg and the New York Rangers as the only teams without a victory this season.
Celebrity Families of 2014 [PHOTOS]