Celtic announced the appointment of Lawwell as head of first team scouting and recruitment on Monday night.
Lawwell was head of scouting and recruitment at City Football Group, which boasts Manchester City and Yokohama Marinos among its clubs.
Postecoglou told Celtic's website: "I was very keen for Mark to join the club and clearly I am delighted that he has accepted this important role and that we have been able to bring him to Celtic.
"Mark is someone who I worked closely with throughout my four years at Marinos, through which time we developed great trust and an excellent working relationship.
"He knows the way I like to work, the way my teams play and the particular player profile which this requires.
"I know he will be a great addition to our team, leading our first-team scouting and recruitment function and delivering on our key objective of identifying and bringing exciting, talented players to the club."
The new arrival joins Celtic a year after his father, Peter, retired from his role as chief executive, although he remains a director on the football club board but not the PLC board.
Postecoglou said: "We have a great team of people at the club and since joining Celtic I have been hugely impressed by the staff around me who have all worked so hard to deliver success."I have said before that I didn't want to make changes just for the sake of it, but we always aim to build and develop everything we do and in bringing Mark in I know we have made an appointment which will be hugely beneficial for us moving forward."
Nick Hammond left his role as head of football operations 13 months ago but Jay Lefevre has been in place as head of scouting operations since 2019.