The prior Australia manager was known as the Parkhead club's top goal to replace Neil Lennon after discussions with Eddie Howe broke down, citing"motives from his or her Celtic's control".
Postecoglou, 55, arrives from Yokohama F. Marinos - that he contributed into the J League name in 2019.
The Greek-born supervisor won two A-League names with Brisbane Roar until he took control of the Socceroos at 2013.
John Kennedy was in accountable for Celtic because Lennon resigned in February, with the team finishing the year 25 points behind champions Rangers in the Scottish Premiership.
Celtic supported the information of Postecoglou's appointment at an announcement to the Stock Exchange shortly before 9am that read:"Celtic is very happy to announce the appointment of Ange Postecoglou since the club first-team director with Ange starting employment with immediate impact.
"The Board of Celtic is very happy to welcome Ange into the team and looks forward to working with him later on."
On his appointment, Postecoglou said:"The chance that's been given to me personally is among the best honours in soccer and also the obligation to direct our glorious football club to the future is just one which I will cherish .
"Celtic is among the names in world soccer, of that there's not any uncertainty - that a giant of a team, a correct footballing association and a whole lot more - history, real material, actual authenticity and actual soul.
"I understand Celtic is an actual way of life for a lot of people and I understand the requirements which include this place - I'm prepared to do everything I can to fulfill those requirements.
"I will do everything I could to receive our amazing club on top and, at precisely the exact same time, produce the type of soccer which our lovers enjoy.
"we would like to entertain our fans and we would like to win, these are the goals which I constantly set myself and that I now start work on.