The €20 million summer signing is now free to make his debut for his new club and could take to the field against Queens Park Rangers after the international break
The Argentina international joined the club for €20 million from Sporting Lisbon earlier this summer but has yet to feature for Louis van Gaal's side, missing all four of the club's competitive fixtures so far this season.
But Rojo is now available for selection and will be able to make his debut in the club's clash with QPR at Old Trafford on September 14.
Rojo, who featured for Argentina in their 4-2 win over Germany on Wednesday, could make his United bow alongside Daley Blind and Radamel Falcao, who both joined the club on transfer deadline day.
It is understood that the hold-up in the 24-year-old receiving his work permit was because of an ongoing criminal investigation in his homeland in 2010, though Rojo reportedly denies any wrongdoing.