The Japanese midfielder has returned to the Bundesliga club on a four-year deal.
The Bundesliga side announced the deal on Sunday afternoon, bringing the Japanese international's spell at Old Trafford to a close.
"The chance presented itself over the last few days to make the signing of Shinji Kagawa from Manchester United," Dortmund sporting director Michael Zorc said.
"Of course we chose to sign this quality player again."
"I said on the day of my departure that my story with Borussia Dortmund was not completed," Kagawa told Dortmund's official website.
"I wanted to realise my Premier League dream but now I'm happy to be back in Dortmund; in this great team, the unique environment. Dortmund is like a family.
"I am proud they have never forgotten me and I can belong again."
Kagawa was signed from Jurgen Klopp's side by Sir Alex Ferguson in 2012, and the midfielder scored a hat-trick in his debut season against Norwich.
But he failed to shine under David Moyes last season, with the Scot struggling to utilise the player to the best of his abilities.
And the writing seemed on the wall for Kagawa's time at United when Louis van Gaal revealed that the player was not interested in adapting to his philosophy.
The coach said: "Kagawa did not wish to fulfil my wishes, my philosophy when I played him in the United States.
"Kagawa was a No.10 at [Borussia] Dortmund but I want to try him at No. 6 and No. 8."
