Postecoglou's side failed to lay a glove on Fulham as the Cottagers ran out 2-0 winners on Sunday.
Rodrigo Muniz broke the deadlock in the 78th minute before former Tottenham defender Ryan Sessegnon added the second ten minutes later.
It was Tottenham's 15th league defeat this season, their equal-most since the 2008/09 campaign.
One fan at Craven Cottage took the opportunity to stick the boot into Postecoglou following the contest.
Instead of letting the fan's swipe wash over him, Postecoglou hit back.
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As Postecoglou made his way to the dressing room, the fan said: "What was that? What was that? What was that?"
The Tottenham boss stopped dead in his tracks and stared down the fan, as Djed Spence and Mathys Tel tried to lead him away.
Postecoglou told the fan to 'behave' and made a calming gesture with his hand.
The Australian added: "You're a young man."
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Tel also tried to intervene and said: "Hey, respect. Respect."
The fan continued to take aim at Postecoglou, which prompted the Aussie to reply: "Behave yourself."
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Postecoglou then moved away as a Tottenham staff member questioned why the fan wanted to poke fun at the evidently frustrated manager.
The former Celtic manager's reaction was not the only clash involving a fan and a Tottenham figure on Sunday.
Tel, who only joined Spurs on loan in January, was captured talking to some of the away fans following the defeat.
Several supporters could be heard shouting, 'It's not good enough'.
But Tel tried to keep the fans onside and responded: "We need to be together."
As for Postecoglou, the incident at Craven Cottage was not the first time he had clashed with a fan.
Following a brutal 1-0 loss to Leicester City in January, which remains the Foxes' only win from their last 14 games, a Tottenham fan told Postecoglou: "You're taking us down, we're going down with you."
Postecoglou appeared to hear the fan's dig as he backtracked up the tunnel and looked to make eye contact with the fan before disappearing down the tunnel again.
The former Socceroos boss also copped an earful from Tottenham supporters following a 1-0 loss away to Bournemouth, where some labelled him 'Postenoclue'.
"They're disappointed and I got some pretty direct feedback, which is always healthy mate," Postecoglou said after the match.
Postecoglou will look to get Tottenham supporters back onside when Premier League resumes following the international break as they travel to Chelsea on April 3.