No details of the fee have been released but German media reported earlier this month that the 23-year-old would cost the Premier League club 17 million euros.
"Shinji is an exciting young midfielder with great skill, vision and a good eye for goal," manager Sir Alex Ferguson told his club's website on Friday.
"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."
Kagawa attracted few headlines when he joined Dortmund for 350,000 euros from then Japanese second division club Cerezo Osaka in 2010 but he surprised many by notching 21 goals in 49 league games in Germany.
He won two consecutive Bundesliga titles as well as the German Cup last season, helping Dortmund establish themselves as the new force in German football.
Kagawa's creativity and darting runs from the middle of the field will bring an added dimension to a United side that brought 37-year-old Paul Scholes out of retirement last season to help their Premier League title challenge.
"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."
United launch their season with an away game at Everton on August 18.
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