Postecoglou's team sit second in the table after going unbeaten in the Australian's first seven Premier League games in charge.
Tottenham left it late to end Liverpool's own unbeaten start as Joel Matip's stoppage-time own goal capped a controversial encounter in north London.
Son Heung-Min put Tottenham ahead after Liverpool midfielder Curtis Jones was sent off when his yellow card for a foul on Yves Bissouma was upgraded to a red.
Liverpool also had a Luis Diaz goal wrongly disallowed for offside by VAR, with referees' chiefs later confirming it was a "significant human error".
Reds forward Diogo Jota was dismissed in the second half for two bookings and Matip's own goal from Pedro Porro's cross sealed Tottenham's victory.
Postecoglou's impact on the club has been remarkable since his close-season arrival from Celtic.
He has lifted the gloom left by his predecessor Antonio Conte's troubled reign and believes the manner of their win against Liverpool was another significant statement after beating Manchester United and drawing at Arsenal.
"Particularly the manner in which it happened, it leaves the real impact and impression on everyone involved," Postecoglou said.
"Another challenge for us today. We had to cope with a fair few things and show a different side of our game today. Second half I thought we were really good.
"It's important. We're still a team in its infancy in terms of the way we want to play, the age and experience of the group."
Tottenham enjoyed more than their share of good fortune on route to victory, but Postecoglou insisted it was a character-building success.
"Nine men doesn't make it easy. Liverpool have been top for a while, have a top mentality and are still dangerous on the counter-attack," he said.
"The disallowed goal was not my decision. Liverpool will be disappointed with it but that's nothing to do with me or us.
"We've had a couple (of own goals) ourselves this year so that's part of football.
"We probably didn't get into the right areas enough, but that last cross I thought was a quality delivery into an area no defender likes to deal with."
