Drogba sustained the injury in Friday's friendly against Japan following a heavy challenge from opposing central defender Tulio and was immediately sent for an operation in Bern.
Ivory Coast Federation spokesman Roger Kouassi announced today the surgery was a success - although a time frame for the striker's return remains sketchy, with the Elephants set to open their World Cup campaign against Portugal on June 15.
Kouassi said: "His recovery has been very encouraging.
"It is suggested that, at this point, all signs are good and there are many hopes."
Drogba is key to Ivory Coast's hopes of progressing out of Group G, which also includes Brazil, North Korea and Portugal.
The African Player of the Year scored 37 goals in all competitions for Chelsea last season and his absence would be a major blow to coach Sven-Goran Eriksson.