Pizzi will lead Saudi Arabia at the World Cup, having resigned as Chile coach after failing to take the two-time defending Copa America champions to Russia.
Argentinian Pizzi takes over a Saudi team that Dutchman Bert van Marwijk led to World Cup qualification. Van Marwijk left in September over a contract dispute.
Bauza took over but his short stay was ended due to poor friendly results.
“We wanted Pizzi because of his achievements in the past and his ambitions for the future,” Saudi Arabian Football Federation president Abdel bin Mohammed Ezzat said in a statement.