The Greek-born coach was unveiled as Neil Lennon’s successor on Thursday morning following weeks of speculation. His apparent farewell to supporters of Yokohama F. Marinos earlier this week fuelled those rumours and the 55-year-old was appointed on a 12-month rolling contract.
Postecoglou, who is taking up his first coaching job in Europe, said: “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.”
McKay, who declared himself “delighted" with the appointment, revealed Postecoglou's vision had been a big selling point.
“In all our discussions, I have been hugely impressed by Ange and his aligned vision to work with me to develop and evolve the club, focused on bringing back sustained success,” McKay told Celtic's official website.
"He is someone I believe has the coaching and managerial credentials to be a real success at Celtic and we are already working closely together on our plans for the season and seasons ahead.
“Throughout all our conversations I have been struck by Ange’s passion for Celtic and his strength of feeling for Celtic, he is hugely motivated by this opportunity, honoured to be Celtic manager and eager to be part of the club’s future success.
“Our discussions have also outlined the exciting vision he has for the team. He is someone who wants to play football the way all Celtic supporters love, an ambition very close to his heart.
“Ange is also very well-connected in global football; he has contacts which I am sure we will benefit from, he has a modern outlook and he shares my desire to evolve the club positively for the seasons ahead.
“Importantly, he is also someone we believe who has the mentality and strength of character to lead a club of Celtic’s size and stature.”
Long-serving Celtic CEO Peter Lawwell, who is stepping aside for McKay, added: “Ange is a manager we have known about for a number of years and someone who has been prominent in our thoughts for some time.
“He is a great coach who has attractive, winning football in his DNA. He has an excellent record, he is a very strong personality and someone with a real desire and passion to take Celtic forward.
“I wish Ange and Dominic all the very best for the future and I am sure they will work towards delivering great success to the club and our supporters.”
Celtic chairman Ian Bankier said: “Ange is a quality manager with an excellent record and someone whom we believe has the ability, desire and passion to deliver success to our supporters.
“He has huge ambition and a passion to be the best in everything he does – his real desire to lead Celtic as manager is very clear.
“We will give Ange every support as we look to prepare for the coming season and beyond.”