Landeskog looked back for a pass near the blueline early in the third period, then turned and was hit hard by Sutton, whose shoulder made direct contact with Landeskog's head.
Sutton was whistled for an elbowing minor on the play. Shanahan explained the suspension in a video, saying the defenseman took an improper route to Landeskog's head and did not deliver a full body check.
"Instead, he picks Landeskog's head," Shanahan says in the video.
Shanahan said it was a clear violation of the NHL's illegal check to the head rule that says: "A hit resulting in contact with an opponent's head, where the head is targeted and the principal point of contact, is not permitted."
Sutton is suspended without pay and will forfeit a little more than $57,432 in salary, which goes into the Players' Emergency Assistance Fund.
He missed Sunday's game against St. Louis and will also miss games against Los Angeles, Phoenix, Montreal and Boston. He is eligible to return November 11 at Detroit.