The Warriors have paid an undisclosed fee for the 25-year-old, who had two years left on his deal with the Robins, to fill the boots of Finch who is returning to Australia at the end of the season.
Wigan coach Shaun Wane said: "I am really excited to have Blake Green joining us. We have admired his style of play for some time at Wigan with his ability to attack the line, his kicking game and his strong defence."
He added: "He will be a great addition to our squad and has all the qualities we need both as a player and a person."
Hull KR have already secured a replacement for Green in fellow Australian Travis Burns and say they have received "substantial" compensation.
Rovers chief executive Mike Smith said: "Blake is a very good player and a consummate professional and we're very sorry to see him move on.
"He went through a lot this season with illness and injury but kept smiling and remained an integral part of the squad, even when he wasn't playing. We'll miss having him at Hull KR and wish him all the very best for the future."
Green, a former Australian Schoolboys captain who joined Hull KR from Canterbury Bulldogs two years ago, was player of the year in his first season after scoring 12 tries in 27 appearances but has played just 10 matches this year due to injury and illness.
Green said: "I'm really excited to be given the opportunity to play for the Wigan Warriors. They are a great club with a proud history which I'm sure will only improve my game. I'd like to thank Neil Hudgell and Hull KR for the way they have treated me over the last two years.
"The fans have been brilliant with me from the day I arrived and I leave Rovers with only great memories and some close friendships. I'm obviously moving onto a new chapter in my career with Wigan and the first two fixtures I'll be looking out for will be the ones against Hull KR."