Kagawa passed a medical with the Old Trafford outfit on Friday and has 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."
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.
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.
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."