The Brazilian has signed a five-year deal which includes a club option for a further 12 months upon its expiry.
The 25-year-old, who is United’s second signing of the summer transfer window after Diogo Dalot, was a target of Manchester City until the Premier League champions withdrew their interest in January.
The eight-cap Brazil international, who is currently on World Cup 2018 duty, underwent a medical at United’s AON Training Complex less than 24 hours after turning out for the Selecao in their pre-tournament friendly win over Croatia at Anfield.
United manager Jose Mourinho told the club's official website of Fred’s capture: “Fred will complement our other midfielders' qualities, which we need; his creative brain and passing vision will give us another dimension to our game. I am very happy he is joining our club and I believe our players know how important he can be in our team.”
The midfielder himself added: “This is the biggest club in the world and I am delighted to be a part of this amazing team. To work with Jose Mourinho, who has won so many trophies in his time as a coach, is an opportunity not to be missed.
”I cannot wait to get started and to meet my new team-mates. I would like to thank everyone at Shakhtar Donetsk for everything they have done for me over the past five years.
“This is a very exciting time for me with my focus now fully on the World Cup with Brazil.”
United’s quest for trophies in 2017-18 was often compromised by their lack of stability in midfield, with Ander Herrera, Marouane Fellaini and Scott McTominay all finding it difficult to pin down a regular starting shirt.
And Fred looks set to become Mourinho’s preferred option in midfield once he joins up with the United squad after representing his country at the World Cup in Russia.
Fred’s transfer fee makes him the fourth most expensive signing in the history of the Old Trafford club, ranking behind only Angel di Maria, Romelu Lukaku and Paul Pogba in United’s list of big-money buys.