The outgoing Hoops chief executive was forced to put together a hurried approach for the Yokohama F. Marinos boss when long-term number one candidate Eddie Howe backed out of a move to Celtic Park late last month.
Having wasted more than three months trying to tempt the former Bournemouth boss north, Celtic have now moved swiftly to bring in the Greek-born Australian.
And Lawwell fully believes that by appointing the 55-year-old in his final major act as Parkhead supremo will ensure the club returns to winning ways.
Lawwell, who stands down after 17 years in charge at the end of the month, said: "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 (McKay) all the very best for the future and I'm sure they will work towards delivering great success to the club and our supporters."
It will be left to incoming chief executive Dominic McKay to work with Postecoglou to forge the route forward.
And it will be no easy task, with a major squad overhaul required and just six weeks remaining before the club are due to kick off their Champions League qualifying bid on July 20.
But McKay is convinced the club have sourced the right man to lead the rebuild.
He said: "I am delighted we have been able to secure Ange as the new Celtic manager.
"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.
"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, contacts which I'm 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 has the mentality and strength of character to lead a club of Celtic's size and stature.
"Everyone at Celtic is pleased to welcome Ange and we look forward to working with him to bring our fans the success which their support deserves."
Chairman Ian Bankier also welcomed Postecoglou to the club, adding: "He 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 and we will be doing all we can to work with him to bring our supporters as much success as possible."