The Argentine defender, 24, joined from Sporting Lisbon for £16 million in August, but was unable to feature after initially failing to obtain a required working visa.
He had to travel to Madrid for an interview regarding his permit because of an alleged altercation with a neighbour in his homeland in 2010.
United have endured a tricky start to the season, accumulating just two points from three league matches and crashing out of the League Cup following a hapless 4-0 defeat to MK Dons of League One.
"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 shortly after his switch to Old Trafford .
"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."
Rojo featured in Argentina’s 4-2 win over Germany on Wednesday night. United’s record signing Angel di Maria bagged three assists and a goal in the friendly, which was a re-run of the World Cup final.
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