The Chelsea striker suffered the injury playing against Japan on Saturday and left the squad for an operation in Bern, Switzerland.
Drogba netted 37 times for his club Chelsea in the Premier League last season and is key to Sven-Goran Eriksson's hopes of taking the Ivory Coast to the latter stages of the first World Cup to be held on African soil.
It is unclear if he could be available for the Elephants opening game against Portugal in Port Elizabeth next Tuesday.
Roger Kouassi, a spokesman for the team, suggested the operation was a success.
"His recovery has been very encouraging. It is suggested that, at this point, all signs are good and there are many hopes," he said.