The Serie A side decided to send the 27-year-old for the procedure to sort an issue from a problem he picked up eight years ago.
Zuniga missed the 2-0 defeat to table-toppers Roma on Friday, and will have to wear a knee guard for the next ten days before it will be decided how long he will require for rehabilitation.
Elsewhere, centre-back Miguel Britos has avoided surgery on a dislocated shoulder.
The Uruguayan sustained the injury in Friday's fixture with Roma, and was replaced in first-half stoppage time to be replaced by Paolo Cannavaro.
He will follow a special training plan at the club, and will be missing for up to four weeks.
Rafael Benitez's charges lie second in Serie A with 19 points from eight matches, and will look to bounce back from their Roma setback when they face Marseille in the UEFA Champions League on Tuesday.
