The 24-year-old Argentine was signed for 16 million pounds (USD 26.24 million) last month but missed the first three games of the season.
"The UK Border Agency has agreed to grant #mufc defender @MarcosRojo5 a visa and he will therefore be available to face QPR on 14 September," United said on Twitter.
Rojo, who can play as a left-sided central defender or wing-back, will increase manager Louis Van Gaal's options in a defence that has struggled to adapt to his preferred 5-3-2 formation.
Van Gaal will also have versatile Dutchman Daley Blind available to make his debut against QPR along with left back Luke Shaw, who missed the start of the campaign due to injury, and on-loan Colombia striker Radamel Falcao.
Argentine Angel Di Maria, the club's record signing, could make his first United appearance at Old Trafford along with fellow midfielder Ander Herrera if the Spaniard recovers from injury.

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