Postecoglou - who joined the Glasgow giants from Yokohama F. Marinos - was met with an underwhelming reaction from fans.
Three defeats in six games, including a loss to Rangers, justified the reasoning behind the unhappiness of fans but the Australian soon turned things around.
Steven Gerrard left as Rangers boss in November and Celtic went on a 32-game unbeaten run in the league, defeating their fiercest rivals twice on the way to recording their tenth SPL title in 11 years.
Petrov - who spent seven years at the Hoops and won four league titles - is impressed by the work ethic of Postecoglou.
"Obviously the uncertainty around the club, especially when Neil Lennon left, there was a couple people on the high level on the board (that) had left as well so there was a lot of uncertainty," Petrov told Planet Sport.
"There was no direction for the club, there was a lot of questions out there. I think the main candidate Eddie Howe was the one everyone was pushing for, everyone wanted him the become the manager - everyone was actually expecting him to takeover.
"But all of a sudden Celtic had made a U-turn. They went to somebody else and they appointed a very interesting man in charge - a man who maybe mostly a lot of people in Scotland don't know about.
"Obviously I know a lot about him because he managed in Japan where I got a lot of friends who played in Japan. They know a lot about Ange Postecoglou so when that appointment came, they were actually quite excited to see how he was going to do in Celtic.
"When I had the conversation about Ange Postecoglou, (I asked) what kind of manager he is, what kind of profile of a manager is he? They were actually very positive about his way of managing.
"His style of play, soon within a couple months I could see why everybody was so happy and so positive about him. Celtic have been able to appoint the man to know exactly what he wanted. He's very clear in his philosophy.
"He is recruiting incredibly good. I think he has completely changed the full team. I think he has signed about probably 12 players and he had three on loan. This is a completely changed transition to a full team," said Petrov.
"So I think in general skill, he has done an incredible job by believing in what he can do, what kind of manager he is. But mainly he had to convince everybody about his style.
"He had to convince his staff and players that style and the way he wants the team to play is the right way by getting the right results, he managed to do that.
"He had to convince the fans, especially with the attractive style, sometimes in certain games they'll control the game, and they'll play well but they won't score enough goals so the fans were asking questions and we in the studio were asking questions.
"He's managed to convince everyone that that's the way - obviously he had to convince the board that his training methods and his way of working is a successful one."