Verhaeghe made it 2-0 at 17:29, converting a centering pass by Gustav Forsling.
"We had some moments early but they scored easy, as far as some of their goals, and they got rolling there and then we were chasing," Blue Jackets coach Brad Larsen said. "It wasn't a good recipe.
"They were a better team. They built a lead and then you get a little scrambly. It wasn't pretty."
Verhaeghe extended the lead to 3-0 at 18 seconds of the second period, scoring off a rebound of an Aaron Ekblad shot. His third goal made it 4-0 at 6:34 when he scored with a snap shot from the left face-off circle.
"I'm playing with two good players (Barkov and Duclair)," Verhaeghe said. "They found me all over the ice. I shot the puck, it goes in a couple of times. It happens."
Nick Cousins made it 5-0 at 8:34 of the third period. After Verhaeghe's fourth goal, Eric Staal scored at 14:16 for the 7-0 final.
"It feels pretty good," Lyon said of the shutout in his 34th NHL game. "I wish I could be super humble, but it actually feels really good. It's one of those things I've always wanted, but the opportunity never presented itself.
"The shutout was nice personally, but I would have been happy winning 7-6. The focus this time of the year is winning."