Argentina defender Rojo was confirmed as a United player last month in a move that saw winger Nani rejoin Sporting Lisbon on loan until the end of the season.
However, Rojo was unable to feature for his new club prior to the international break, with manager Louis van Gaal expressing his surprise at the delay.
A statement from United on Thursday read: "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 September 14.
"Rojo completed his move to Old Trafford from Sporting Lisbon on August 20 but the requirement to obtain a visa has thus far prevented the 24-year-old from making his debut.
"There was no such delay for fellow Argentinian arrival Angel Di Maria, who already has an Italian passport as one of his grandparents is from Italy."
Rojo is one of six new faces to have been brought to the club by Van Gaal, along with Di Maria, Ander Herrera, Luke Shaw, Daley Blind and Radamel Falcao.
A left-sided player who can feature at either centre-back or left-back, it remains to be seen where Rojo will be deployed by Van Gaal as the Dutchman continues to try and find a system which best suits his players.
The arrival of Blind from Ajax on transfer deadline day, as well as that of Rojo, looks to have gone some way to addressing some of United's frailties, and the club's fans could see the Argentinian in Premier League action at Old Trafford a week on Sunday.
