The 27-year-old joins the English Premier League giants following a three-year spell with Dortmund, where he made 140 appearances and scored 41 goals.
"I am very proud to join Manchester United, this move is a dream come true for me," Mkhitaryan told the club's official website.
"I am excited to play for a club with such an illustrious history and hope to be part of it for a long time. I thank the trust the club and Jose Mourinho have put in me.
"Finally, I believe playing for such a great club honours my father’s memory, and the inspiration and drive he gave to me when I was young."
Newly-appointed United boss Jose Mourinho added: "Henrikh is a very talented footballer who has been in such prolific form for both his club and his country.
"He is a real team player with great skill, vision and also has a good eye for goal. I am delighted he has chosen to sign for United.
"I believe he will make an impact on the team very quickly as his style of play is suited to the Premier League. We are all looking forward to working with him."
While United have not revealed the financial details of the transfer, the Armenia international is believed to have penned a deal worth £200,000-a-week.
Several press outlets in England have also suggested the transfer fee is worth at least £80-million, with the player receiving a £6.7m signing-on fee.