The 24-year-old Argentina defender joined United from Sporting Lisbon on August 20 for a reported fee of £16 million ($26.3 million, 20.2 million euros).
But his visa application was delayed while British immigration officials investigated an alleged altercation that he had with a neighbour while living in Argentina in 2010, which resulted in criminal charges.
However, United announced on their website: “The UK Border Agency has agreed to grant new Manchester United signing Marcos Rojo a visa and the defender will therefore be available for the Reds’ next game against QPR on 14 September.”
Rojo’s fellow Argentina international Angel di Maria, who has also joined United, did not experience a delay in his visa application because he already possesses an Italian passport.
Rojo qualified automatically for a work permit because he has played in more than 75 percent of Argentina’s competitive fixtures over the last two years.
Both Rojo and Di Maria played in Argentina’s 4-2 friendly win over world champions Germany in Duesseldorf on Wednesday, in which Di Maria set up three goals and scored another.