The Red Devils have paid the Portuguese club around £16m for the versatile Argentina international, and the defender will wear the number five shirt at Old Trafford.
Rojo, who played in the World Cup final, can operate at both left-back and centre-half and signed a five-year deal with Louis van Gaal's side.
Meanwhile Nani, who signed a five-year contract extension 11 months ago, will rejoin his former club on a season-long loan deal.
Rojo told the club's official website: "It is such an honour to say that I now play for Manchester United. The Premier League is the most exciting league in the world and to have the chance to play in it for the world's biggest club is a dream for me."
"I am young and am very keen to continue to learn the game, so playing for such an experienced technical coach as Louis van Gaal is a fantastic opportunity for me.
"I have joined United to work hard with my team-mates to win trophies and I know that the manager shares that ambition."