The Argentina international will cost United just under £16million and moves to Old Trafford on a five-year deal.
"Manchester United is delighted to announce the signing of defender Marcos Rojo from Sporting Lisbon for a fee of 20million euros, with Nani joining the Portuguese club on a season-long loan," said a club statement. "Marcos joins on a five-year contract."
Rojo is equally comfortable at centre-back as he is at left-back and played all 120 minutes of July's World Cup final as Argentina lost to Germany after extra-time in Rio de Janeiro.
The 24-year-old began his career in his homeland with Estudiantes de la Plata and spent a single season with Spartak Moscow before moving to Sporting in 2012.
He also has 28 caps for his country and cannot wait to play his part in Louis van Gaal's new era at United.
"It is such an honour to say that I now play for Manchester United," he said.
"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."
Rojo joins Ander Herrera and Luke Shaw as Van Gaal's third high-profile signing since taking over earlier in the summer and the Dutchman believes he will be a useful addition as the former Bayern Munich boss implements a new 3-5-2 formation at Old Trafford this term.
"Marcos is a very gifted defender," he said. "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."
United lost centre-backs Rio Ferdinand and Nemanja Vidic alongside left-back Patrice Evra this summer and Rojo will add to a defensive unit which was largely criticised following the defeat to Swansea at the weekend.