Louis van Gaal's patched-up team crashed out of the Champions League on Tuesday when they lost 3-2 in Wolfsburg.
Rojo, along with six other first-team players including captain Wayne Rooney, missed the match at the Volkswagen Arena.
Last week, the Argentina defender dislocated his left shoulder after a bad fall in training
It was the same injury he suffered in the 1-0 defeat at Manchester City last season.
Van Gaal hoped Rojo would not require surgery, but the defender confirmed on Wednesday morning he had gone under the knife.
Alongside a picture of him in a hospital bed, he wrote on Instagram: "My operation was a success
Thank you to my beautiful wife, always with me
Thank you everyone for your messages of love and support."
Van Gaal is likely to reveal for how long he will be without Rojo at his pre-match press conference on Friday.
The Dutchman was without Rojo, Rooney (ankle), Morgan Schneiderlin (hip), Ander Herrera (hamstring), Phil Jones (ankle), Luke Shaw (broken leg) and Antonio Valencia (foot) for the loss in Germany.
To make matters worse, Matteo Darmian pulled his hamstring in Wolfsburg and Chris Smalling suffered a groin strain in the defeat
Both are likely to miss Saturday's match at Bournemouth.
Bastian Schweinsteiger is also facing a possible three-match ban for an alleged elbow on Winston Reid in United's 0-0 draw versus West Ham last weekend
The German has until noon on Thursday to decide whether he wants to contest the Football Association's charge of violent conduct.
Source : PA
Source: PA