The club remain confident that Johnson will recover in time to play in pre-season training during the summer.
Johnson missed Wednesday's match as Fulham ended a run of five successive defeats with a 1-0 win over Portsmouth.
The former Everton forward has had a miserable few months spending lengthy spells on the treatment table.
He suffered a shoulder injury at the start of the season and then had groin problems in November.
Although AJ has relatively done nothing this season its still a big loss for us Brede_Butter_Pudding Johnson was in Colorado in the United States for Thursday's operation conducted by Dr Richard Steadman, the surgeon who also worked with Manchester United's Owen Hargreaves and golfer Tiger Woods.
Fulham boss Roy Hodgson said earlier in the week: "I was still believing he was going to come back and it would not be that long."
Johnson, who joined Fulham from Everton for £10.5m in August 2008, played the full 90 minutes for the Cottagers in their 2-0 defeat by Blackburn on 17 January, giving Hodgson renewed hope his forward was back to full fitness.
Hodgson also revealed that Johnson was feeling "shattered" and "devastated" ahead of the operation.
Fulham are already without injured forwards Clint Dempsey and Zoltan Gera, with Hodgson admitting he may regret loaning Diomansy Kamara to Celtic.
Injury crisis worst ever - Hodgson
"Would I like Kamara back here? I would but it's too late," added Hodgson.
Kamara signed a loan deal until the end of the season with the Scottish club on transfer deadline day - before Hodgson knew the severity of Johnson's latest setback.
"One of the reasons why we let Diomansy go was because we didn't think the problem with Andy was anywhere near as grave," said Hodgson, who still has regular strikers Bobby Zamora and Erik Nevland to call on.
"I was concerned that I would not be able to give Diomansy regular games, so when a club of Celtic's stature comes in for him it is very difficult for me to say 'you have to stay'.
"Of course, now I have let him go and have further news about Andy, I might be beginning to regret it but you can only make decisions at the time."
Fulham's injury list has grown, with United States international Dempsey suffering knee ligament damage and Hungarian Gera taking a blow to his knee in training last week.
Hodgson did sign Roma striker Stefano Okaka on loan until the end of the season, however and the Italy Under-21 international featured for the first time for Fulham against Portsmouth.