Mkhitaryan is new United manager Jose Mourinho's third signing, after deals for Eric Bailly and Zlatan Ibrahimovic.
The 27-year-old scored 41 goals in 140 games in a three-year spell at Dortmund and was voted Bundesliga Players' Player of the Season after netting 11 goals last term.
Mkhitaryan is new United manager Jose Mourinho's third signing, after deals for Ivory Coast defender Eric Bailly and ex-Sweden captain Zlatan Ibrahimovic, as the Portuguese rebuilds the FA Cup winners after a disappointing league campaign.
"I am very proud to join Manchester United, this move is a dream come true for me. I am excited to play for a club with such an illustrious history and hope to be part of it for a long time," Mkhitaryan told the club's website (www.manutd.com).
Aremenia captain Mkhitaryan is his country's record scorer with 19 goals in 59 matches since making his international debut in 2007.
Although the 13-times Premier League winners, who missed out on a Champions League spot after finishing fifth last season, did not reveal any financial details, British media estimated the transfer fee at 30 million pounds ($38.62 million).
Calm and composed under pressure, Mkhitaryan can break open the tightest defence with his passing skills and reading of the game and Mourinho felt the Armenian would adapt seamlessly to the physicality of English football.
"Henrikh is a very talented footballer who has been in such prolific form for both his club and his country," Mourinho said.
"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."
Despite Dortmund's reluctance to sell, chief executive Hans-Joachim Watzke said last week that United's offer was too tempting to reject, especially with the player entering the final year of his contract.
"Manchester made us an enormously lucrative offer Had we rejected it the player would have left in 2017 without a fee and we would have just delayed the issue for a year," Watzke said in a statement.