The Greek-born Australian has left his role as Yokohama F. Marinos boss after three years to join the Scottish giants, who parted ways with Neil Lennon in February.
The 55-year-old will be tasked with leading Celtic back to glory in the Scottish Premiership after their decade of dominance was put to an end by Rangers last season.
Speaking following his confirmation as the new Bhoys boss, Postecoglou told Celtic's website: "The opportunity that has been given to me is one of the greatest honours in football and the responsibility to lead our magnificent football club into the future is one that I will cherish dearly.
"Celtic is one of THE names in world football, of that there is no doubt - a giant of a club, a proper footballing institution and so much more - real history, real substance, real authenticity and real soul. I know Celtic is a true way of life for so many people and I know the demands which come with this position - I am ready to do all I can to meet those demands.
"I will be doing everything I can to get our great Club back on top and, at the same time, deliver the kind of football which our fans appreciate. We want to entertain our fans and we want to win, these are the objectives which I always set myself and which I now begin work on.
"When you think of Celtic, you think of supporters and my dream is to see our fans back at Celtic Park with us as soon as possible. We all hope things are changing for the better and can see our fans soon as they are vital to everything I want to do. We need our supporters back by our side and I can't wait to be with them back in a packed Paradise.
"I have already had great discussions with Peter, Dom and the Board about their ideas and strategy for the future of the club. I know the Club's new modern vision aligns very much with mine and we now look to go and deliver on this. In everything we do we aim to give our fans a successful team of real quality, which they are excited about and can be proud of.
"We have already begun work on our plans for adding to the squad - we aim to bring players of quality to Celtic to enhance the existing core of great talent.
"I very much look forward to meeting with the players on their return from the close season break and I can't wait to get started as Celtic Manager."
Coronavirus rules mean the 55-year-old will isolate once he touches down in Scotland from Japan on Thursday, which sits on the UK Government's amber travel list.
This means Postecoglou is unlikely to be at the club's Lennoxtown training base on June 17 when the first-team squad report back for the start of their pre-season preparations.
Celtic are also waiting to hear back from UEFA after applying for an exemption on rules requiring bosses to possess a Pro-Licence - which Postecoglou does not hold - if they wish to manage in European competition.
After Lennon resigned back on February 23, Celtic spent weeks chasing number one target Eddie Howe.
But when their move for the former Bournemouth boss broke down, they were forced to switch their attentions to Postecoglou.
The former Australia boss has won league titles in his homeland and in Japan, while he also led the Socceroos at the 2014 World Cup before resigning just weeks ahead of the 2018 finals in Russia.