The Japanese playmaker has signed a long-term contract with the Old Trafford club after successfully undergoing a medical and gaining a UK work permit
The Old Trafford side announced earlier this month that they had reached an agreement with Borussia Dortmund over the transfer of Kagawa.
United have now announced that the Japanese international has passed a medical and obtained a UK work permit.
Sir Alex Ferguson has spoken of his delight that the 23-year-old is on his way to the Red Devils.
"Shinji is an exciting young midfielder with great skill, vision and a good eye for goal," he told ManUtd.com.
"I am delighted he has chosen to come to United. I believe he will make an impact upon the team very quickly as he is suited to United's style of play.
"We are all looking forward to working with him."
Kagawa has also spoken of his excitement at getting started.
"This is a challenge I am really looking forward to," Kagawa told ManUtd.com.
"The Premier League is the best league in the world and Manchester United is such a massive club.
"This is a really exciting time for me and I can't wait to meet the team and start playing."