Cooke was issued a five-minute major for charging for the hit on Tyutin, which sent him face-first into the glass.
Cooke also was handed a five-minute major for fighting, as was Columbus center Derick Brassard, who went after Cooke after the hit. Tyutin stayed in the game.
The Penguins, who lost 4-1, contended Tyutin had glanced over his shoulder before playing the puck behind his net.
"It was the worst hit I've ever had from behind," Tyutin said, according to the Columbus Dispatch. "I wasn't surprised, not when you see [Cooke] in the highlights all the time for dirty hits."
Tyutin knew the hit was coming and looked Cooke "right in the eyes," Penguins coach Dan Bylsma said, according to the Post-Gazette.
"It used to be you were responsible for yourself if you turned, but that's not the way anymore," Cooke said, according to the Pittsburgh Post-Gazette. "That call's been called on us four or five times this year. It's the same thing.
"He turns and looks and sees me coming. The other defenseman is slowing me down, so I'm not skating full speed," he said.
Although the four-game suspension was for the hit on Tyutin, Cooke was at the center of another questionable hit earlier in the week.
On Sunday, late in the third period of the Penguins' nationally televised loss at Washington, Cooke made knee-on-knee contact with Capitals star Alex Ovechkin. Cooke was given two minutes for tripping on the play.
Neither Ovechkin nor teammate Nicklas Backstrom, who both charged at Cooke and landed punches, was assessed a penalty for his response to the hit.
Bylsma also defended Cooke Sunday for the hit on Ovechkin. But Capitals coach Bruce Boudreau wasn't buying it.
"It's Matt Cooke. Need we say more?" Boudreau said Sunday.
"It's not like it's his first rodeo. He's done it to everybody. And then he goes to the ref and says 'What did I do?' He knows what he did. There's no doubt in my mind. He's good at it."
Although Savard returned for the playoffs, he missed the first 23 games of this season with post-concussion syndrome. He has been ruled out for the season after sustaining another concussion last month.
Information from The Associated Press was used in this report.