The 30-year-old Uruguay international midfielder was admitted to the Pineta Grande clinic in Castel Volturno after sustaining the injury during training ahead of Sunday's trip to Inter.
Gargano, who started Napoli's past four Serie A matches after sitting on the bench for the previous two, will spend the next four weeks on the sidelines avoiding physical contact.
"The surgery was performed by Professor Giampaolo Tartarus and Dr. Mario Sant'Agata, assisted by Dr. Vincenzo Schiavone, anaesthesiologist, and Dr. Mimmo Falco," Napoli said in a statement.
"The surgery was a complete success. The fracture was reduced and synthesised with a plate and screws.
"Gargano will be able to resume light training on Monday but must avoid physical contact for the next four weeks."
Gargano returned to Napoli at the start of the 2014-15 campaign after spending the past two seasons out on loan at Inter and Parma.