By Brian Homewood
"The necessary surgical intervention has been a success," the Ivory Coast football federation said on its website (www.fif-ci.com) Saturday.
The federation added that medical staff were confident Drogba would make "a rapid recovery" but did not elaborate.
The 32-year-old striker was injured during Friday's 2-0 friendly win over Japan in Sion.
Drogba, a talismanic figure for the Elephants with 43 goals in 67 appearances, limped off after a challenge from Tulio sent him falling awkwardly to the ground after 15 minutes.
If he misses the World Cup it would be a huge blow to the Ivorians, who have been transformed from a middle-ranking team to a major force in African football since Drogba came on to the scene.
Born in Ivory Coast and raised in France, the Chelsea forward has helped them qualify for their first two World Cups in 2006 and 2010.
Friday, vice-captain Kolo Toure said a dejected Drogba had told team mates in the dressing room he was out of the tournament, which starts Friday.
"It's obviously very sad, for him he said that the World Cup is finished, we're just waiting for what the doctor says," Toure said.
But coach Sven-Goran Eriksson said as he left the stadium there was still a chance Drogba could play.
(Editing by Alison Wildey)