The Belgium midfielder underwent the procedure to repair tendon damage at a hospital in the North West over the weekend after a troublesome back complaint failed to clear up as hoped.
"He had his operation on Saturday morning," said United manager David Moyes.
"We just couldn't get him recovered from his back. We thought it would be okay after 10 days but it wasn't getting any better so we made a quick decision to get the wrist done.
"We think it will be six weeks, but there is a chance it could be shorter than that.
"His back problem is going to take time. That is a muscle injury which wasn't healing.
"It is unlucky. He has not felt 100 per cent because of his wrist. He has been finding it difficult.
"I asked him to see if he could get over the Christmas period. He got the back injury against Everton and since then he has never really been right.
"I have no doubt he will eventually be a good player for us."
The £27.5million summer signing from Everton sustained the injury playing against Shakhtar Donetsk in the Champions League in early October but has played since with a protective cast.
If Fellaini's recovery goes as scheduled, he will miss seven Premier League games, including the trip to Chelsea on January 19, plus both legs of the Capital One Cup semi-final with Sunderland and the FA Cup third and fourth rounds.
However, the 26-year-old should be back well ahead of the trip to Olympiakos on February 25 in the first leg of United's Champions League last-16 tie.
Fellaini will miss the Boxing Day trip to Hull, when striker Robin van Persie (thigh) and midfielder Michael Carrick (Achilles) are also set to be missing, although Moyes is refusing to abandon all hope.
"This is the time when miracles happen," he said. "Let us hope there is one for Robin van Persie and Michael Carrick."
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