The former Australia head coach arrived in Glasgow on Wednesday following a quarantine period, which was only a minor delay in relation to the length of time the club took to replace Neil Lennon, who stepped down on February 23.
Celtic had looked set to appoint Eddie Howe following the end of the season but the former Bournemouth manager stunned Hoops fans by pulling out of a move north.
Postecoglou quickly emerged as the new front-runner and the Athens-born Australian dismissed any concerns about not being the first choice after arriving from Yokohama F Marinos.
"You're assuming I was second choice, I might have been fifth choice, you never know," he said at a media conference.
"It doesn't really bother me. What's important is that I have been given the responsibility and opportunity. Just that fact alone shows that the people who made this decision have faith in me.
"That's all I need. From then on it's up to me to get down to working the way I want to and try to bring success to this club.
"Look, I gather there was a little bit of a mixed response when I got announced as the Celtic manager.
"But what I do know is 100 per cent of Celtic supporters will want me to be successful because nothing is more important to them than this football club, and if I am successful then it means this club is successful. I just can't wait to see them back in here.
"My role is to try and provide a club they are going to be proud of, a team they are going to look forward to watching every week, and a team that everyone is talking about.
"If I can achieve those things, and obviously success follows that, then hopefully I have honoured the role I am in.
"But so far everyone has been super supportive and I can't wait to experience it first hand."
Incoming chief executive Dominic McKay, who officially succeeds Peter Lawwell next week, did his best to explain the delay.
"Sometimes to get the best you just have to wait a little bit longer," McKay said.
"I'm utterly delighted we've managed to secure Ange. I've spent time in Australia, spent a lot of time in Japan, so I knew all about Ange.
"When the opportunity came to speak to Ange, it was a great conversation - one of the best, if not the best, I've had in the last five weeks.
"When I offered Ange the role he immediately accepted and I could tell immediately he had the passion, the keenness and the desire to get Celtic back to winning ways.
"We wanted a winner and his experience, being a bit of an outsider like myself, is enormously valuable."
Postecoglou disagreed with suggestions that he was taking a major leap in his career.
The 55-year-old, who led Australia to the 2014 World Cup and won the J-League with Yokohama, said: "I'm not sure what you mean by 'jump' but I'm assuming you are saying I worked at a lower level. I guess that's a matter of opinion.
"I have coached at a World Cup, I have coached against some of the best teams in the world.
"That's not how I look at it. I look at it as another opportunity, an exciting challenge, an opportunity to work somewhere I know people are so passionate about the game.
"With the history and traditions of this football club, and I know how passionate the supporters are, everyone tells me the experience at Celtic Park is like no other. That's why I am here.
"If that means pressure... if I didn't want pressure I would have chosen another occupation, I would have stayed coaching my local club. I am here because this is where I want to be, this is where I want to coach, this is where I want to have success.
"I think most people understand that and I think I will get judged on what happens moving forward rather than what happened in the past."