Sporting Lisbon on Tuesday night sanctioned the sale of the Argentina international to United, with Portuguese winger Nani returning to his former club as part of the deal.
"I'm very happy as this is a big step forward in my career and it's amazing to be joining the biggest club in the world," Rojo told the club's official website.
"I know a lot about the club because I always follow the Premier League and Manchester United were always my team because they were the biggest and the best. Juan Sebastian Veron played for United and was a great player and I always watched his progress. United were always my club.
"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."
Sporting welcomed Nani back to the Estadio Jose Alvalade and revealed that they will use €9 million (£7.2m) from the £16m fee for Rojo towards the construction of their new indoor arena, the Pavilhao.
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.
Van Gaal, whose first league match in charge of United ended in a shock home defeat by Swansea City last Saturday, said: "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."
- Sports & Recreation
- Manchester United
- Juan Sebastian Veron
- Premier League