The financial terms of the deal were not disclosed.
The 23-year-old made his NHL debut with Toronto last season. He had a goal and two assists in 22 games.
D'Amigo has played with the Maple Leafs' AHL affiliate, the Toronto Marlies, since the 2010-11 season. He had 20 goals and 33 points in 51 games during the 2013-14 campaign.
"When I got the call from the GM (Jarmo Kekalainen), it was a reassurance from him that there's a spot for me and if I work hard this summer, I'll have a really good chance. It's one of those things where you're sad to leave but also excited to join a new organization. Not that I did anything bad in Toronto, but it's almost like a fresh start for me.
D-Amigo was a sixth-round pick for Toronto in the 2009 NHL Draft.
Ryan Johansen is the last restricted free agent for Columbus without a new contract.
- Sarah Barshop