United winger Nani - who signed a five-year deal himself at Old Trafford last September - has returned to his former club as part of the deal, switching to the Estadio Jose Alvalade on a season-long loan.
Both clubs confirmed that the protracted transfer had been completed on Tuesday, with statement a released by Sporting revealing that Premier League giants United will make an initial down payment of €10m, followed by further instalments of €5m on December 1 this year and July 1 2015.
The deal had been delayed due to the complexity of third-party ownership, but the issues have now been resolved and the Argentina international defender is delighted to have finalised all the formalities.
"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," the 24-year-old, who caught the eye recently at the World Cup, told United's official website.
"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."
Rojo has become the third signing of the Louis van Gaal era.
The Dutchman enthused: "Marcos is a very gifted defender.
"He has played at the highest level in world football and can play either as a central defender or left-back.
"He has ability, physical strength and a willingness to learn that means he has a very bright future ahead of him. He had a very strong World Cup and has been playing in Europe for a couple of years now. He is a very good addition to the team."
Rojo - capped 28 times for his country - made 61 appearances in total during his two seasons at Sporting, who he joined from Spartak Moscow.
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