Falcao, his country's big hope for their first appearance at finals since 1998, sustained cruciate ligament damage in his left knee while playing for Monaco in a French Cup tie on Wednesday.
"Radamel Falcao's surgery this morning in Porto by Dr Jose Carlos Noronha and his team has been successful," Monaco said on their website (www.asm-fc.com).
Colombian fans fear he will be unable to recover in time for the tournament in Brazil but Portuguese orthopaedic surgeon Noronha did not rule out the 27-year-old, nicknamed Tiger, being fit in time.
"I'm going to exaggerate a little but I think he'll have a 50 percent chance of going," Noronha said of the former River Plate, Porto and Atletico Madrid striker.
"We must go step by step and within three and a half months we'll be able to say something more sure," he told a news conference.
Falcao delivered a message to his fans via Colombia President Juan Manuel Santos who visited him in Porto.
"I think Mr President is the spokesman for 47 million Colombians who were praying to God for the surgery and my recovery. That's a huge spiritual boost for me ... that he should visit me at this difficult time," said the striker.
Falcao will leave Porto for Monaco on Monday to begin physiotherapy, said Colombian team doctor Carlos Ulloa.
The South Americans face Greece, Ivory Coast and Japan in World Cup Group C.
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