Mayfield gave New York a 1-0 lead at 10:34 of the first period when he poked a rebound from Lee, and Dobson made it 2-0 when his shot ricocheted off Kevin Shattenkirk's stick blade into a half-empty net at 12:08.
"It just worked out," Mayfield said. "We played well in the offensive zone."
The lead grew to 3-0 at 5:34 of the second when Jean-Gabriel Pageau sprung Wahlstrom for a semi-breakaway, which the right wing finished by lifting a backhand over Gibson's pads.
Mayfield scored his second of the game at 12:55, a slap shot from the top of the right face-off circle to make it 4-0, and Beauvilier one-timed a Ryan Pulock feed from the left circle 1:32 later at 14:27 to make it 5-0.
"With us turning it into a track meet, it started turning into odd-man rush after odd-man rush, and I certainly wasn't going to have any more of that for John going into the third," Eakins said. "We have an excellent goaltender, we have a world-class goaltender. But we can't just rely on him to do it all himself.
"We've got lots of guys who can score here and make plays, but you can't win in this league unless you're willing to play good-layered defense."