The Japan midfielder has signed a four-year deal after passing a medical with the Old Trafford club on Friday.
United manager Sir Alex Ferguson says he expects the 23-year-old to prove a big success in England.
"Shinji is an exciting young midfielder with great skill, vision and a good eye for goal. I am delighted he has chosen to come to United," said Ferguson.
He added: "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.
Kagawa helped Dortmund win successive German titles, contributing 17 goals and 13 assists as they completed a league and cup double last season.
A 23-year-old playmaker whose 13 goals in 31 games helped Borussia Dortmund win the Bundesliga title. Kagawa was a bargain when he joined from J-League side Cerezo Osaka for just 350,000 euros in 2010
Dortmund total appearances: 56 Goals: 24
The deal is reported to be worth an initial £12m and could reach £17m depending on success and appearances.
Ferguson was present when Kagawa scored in Dortmund's 5-2 win over Bayern Munich in the German Cup final, with the player able to play wide or in a more central role.
And he signalled his intention to get started for the Red Devils.
"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 and I can't wait to meet the team and start playing."
Kagawa has obvious appeal for United's vast Asian fanbase but commercial director Richard Arnold has previously denied any suggestion that players are signed to boost profits.
He said: "Our popularity in certain countries or regions may depend, at least in part, on fielding certain players from those countries or regions.
"We don't sign players to sell shirts. We are reliant on 25 players and they are all massive stars. We have 25 George Clooneys."
Shinji Kagawa has completed his move to Manchester United from Borussia Dortmund for an undisclosed fee.