Spurs twice equalised through captain Heung-min Son either side of half-time and took the game to last season's Premier League runners-up with a high-risk, attacking approach in their head coach's image.
The result leaves both north London clubs still unbeaten this season, and level on 14 points, but Spurs will be the happier of the two after demonstrating that they can match one of the best sides in the country just seven games into Postecoglou's tenure.
Centre-back Micky van de Ven says Spurs already fear no one, thanks to their father figure in charge.
"We don't have fear of another team,” he said. "It would be strange if you have fear for another team. If you go into a game and are like, 'What a team they have', I don’t think it's positive.
"If you have some players who don’t really like to play with the ball and they have a big fear or something, they are going to lose the ball.
"But the trainer [Postecoglou] is telling us, ‘Just keep playing and if you lose the ball it can happen. Just try again. Keep trying, keep trying’.
"If we lose three balls in a row he doesn’t want us to shoot the ball into the stands. We have to just keep playing and have the trust from the trainer.
"As a centre-back, for me it is okay to imagine I lose the ball one time and a big chance comes from it and the trainer is screaming from the sides, ‘Again, again, again.’ It gives us some trust as a player to build up again and again.
“He is really clear. He is really straight. We know what he expects from us. We know what we have to do for him. He is bringing us together like a family.
"He is more like a dad or something. We are like brothers together and we have to do this for each other. You see this on the pitch and we run for each other, we fight for each other and we do everything for each other. He is building that.”
"I think we deserved more because we played great football," said Spurs goalkeeper Guglielmo Vicario, who made fine saves to deny Eddie Nketiah, Gabriel Jesus and Saka. "And we showed everyone who we are. We have to keep improving and working. We take this point but we want to do more.
"We followed the rules of the gaffer. We have to play from the back. We have to take some risks to play this kind of football but it’s exciting. We have to keep working to improve that.
"The gaffer was quite happy but the most important thing was showing everyone who we are. Just this. And I think we showed this because we deserved more."
Six of the visitors' starting XI, including full debutant Brennan Johnson, were experiencing a north London derby for the first time but Son praised the maturity of Spurs' young stars, saying what they lack in experience they make up for in character.
"Look, they know what it means to play in the Premier League," said the Spurs skipper, who has now scored five goals in his last three games.
"Obviously they're missing the experience but they are covering the experience with passion and bravery, which is very, very important.
"I hope we can continue in this way with a very good attitude. Obviously you can't see our training sessions but everyone is working really, really hard to be part of it.
“This performance wasn't just random. If you see our training sessions every single day, it is not normal. We are working really, really hard, intense.
"Everyone is pressuring each other to improve, which is very, very good. If you have a young team with good rhythm, we can exploit this situation and that's what we're looking for."