The 27-year-old Armenia international becomes United's third major summer signing under new coach Jose Mourinho, following on from Zlatan Ibrahimovic and Eric Bailly.
A Dortmund statement read: "After intensive discussions BVB have complied with the wishes of the Armenian national team player to join Manchester United."
Dortmund have not revealed the transfer fee for the former Shakhtar Donetsk playmaker but said United had made an "enormously valuable offer".
Media reports this week suggested United had agreed a deal in the region of £26million for Mkhitaryan, who had 12 months left on his contract with the Bundesliga outfit
Dortmund chief executive Hans-Joachim Watzke said: "Manchester (United) have made us an enormously valuable offer
"Had we rejected it, the player would have been able to leave for free in 2017
And all we would have done is delay the issue for another year."
Source : PA
Source: PA