Chile were cruising after taking a 2-0 lead by the 14th minute but Cahill's header in the 35th made for a tense, end-to-end encounter settled only when substitute Jean Beausejour grabbed a third in stoppage time. (Agencies)
Australia will have to perform miracles to progress from the section after the loss in Cuiaba, with Spain and the Netherlands their other opponents. Reigning champions Spain were thrashed 5-1 by the Dutch in the other Group B encounter on Friday.
"At the start we showed a bit of fear, but when we stepped up and started getting in their faces, we started rolling them a bit," Cahill said in television interview after the match.
"When they showed a bit of aggression, we showed them a bit back. We will learn from this and go into the next game very positively," he said.
Australia next play the Netherlands in Porto Alegre on Wednesday.
"The good thing is that we believe in our football, we believe in our structure. Last week they were questioning whether we could create chances - we did create quite a lot," added the 34-year-old Cahill.
Postecoglou also felt his side had been hesitant early on.
"Maybe we did perhaps give them too much respect and took a while to settle down," he told reporters. "Credit to the players, it would have been easy to give up, but we took the game to them," he said.
He conceded that it would not be easy for Australia to reach the knockout stages, but vowed they would try.
"This a hard group, very finely balanced. You think the Spanish are going to bounce back, but Chile will give them a difficult game. The Dutch will fancy they can take three points off us, but we'll make it as hard as possible," he said.
Cahill's goal was a record-extending 33rd for his country. "It is all about defining moments. I say this all the time, being one of the older boys - this is the stage to do it, when you have to show it," said the forward.
Chile were cruising after taking a 2-0 lead by the 14th minute but Cahill's header in the 35th made for a tense, end-to-end encounter settled only when substitute Jean Beausejour grabbed a third in stoppage time.