Kagawa has passed a medical with the Old Trafford outfit and now tied up the loose ends of a transfer Sir Alex Ferguson expects will prove a big success.
"Shinji is an exciting young midfielder with great skill, vision and a good eye for goal," said Ferguson.
"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 is United's second capture of the summer, following the arrival of England Under-19 international Nick Powell from Crewe.
Ferguson has been under pressure to strengthen his midfield for two years now, with many fans believing it lacks quality compared to a number of United's rivals.
Paul Scholes did abandon his retirement in the middle of last season and, together with Ryan Giggs, has signed on for a further year with the Red Devils.
However, with question marks over the fitness of Anderson and Darren Fletcher, who is still to attempt a comeback after it was revealed he was suffering a bowel complaint, Ferguson had to make a move.
United did miss out on Lille's highly-talented playmaker Eden Hazard, who opted to join Chelsea.
However, Kagawa does bring some impressive statistics to United.
The 23-year-old helped Dortmund land successive German titles and scored 17 times in addition to claiming 13 assists as they also went on to complete the double last season.
Having been ruled out of Olympic duty in London this summer, it is anticipated Kagawa will be part of United's pre-season tour to South Africa and China.
He should also avoid claims of being little more than a signing to sell shirts, a label that was attached to Park Ji-sung for so long.
"This is a challenge I am really looking forward to," said Kagawa.
"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. I can't wait to meet the team and start playing."
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