As in last season's home derby, Spurs were undone by a set-piece on Sunday as Arsenal claimed three successive derby wins at Tottenham for the first time since 1988.
The result left Spurs with just one win from four matches so far this season under former Socceroos boss Postecoglou.
Spurs had started strongly with Dejan Kulusevski denied by David Raya before Dominic Solanke sent a header wide.
Arsenal also had moments on the break with Gabriel Martinelli squandering a good opening not long after Kai Havertz had a header saved by Guglielmo Vicario.
The second period followed a similar pattern initially with a Solanke header deflected wide and Micky van de Ven able to test Raya, but Tottenham's set-piece kryptonite hurt them just after the hour.
After Saka saw a shot blocked by Pedro Porro, his in-swinging delivery from a corner found Gabriel, who brushed past Cristian Romero to head home what proved the winner in the 64th minute.
"I know for some reason people think I don't care about set-pieces and it's a narrative that you can keep going on for ages and ages," Postecoglou said.
"We work on them all the time, like we do for every other team.
"You know that they're a threat. As I said, for the most part, we handled them really well today, but we switched off for one and we paid a price and you learn from that and you move on.
Ange Postecoglou gestures from the touchline during the match against Arsenal. (AP PHOTO)
"It's my burden to carry, mate and I'm happy to do that. For me, there's a bigger picture that's at play here that's much more important than the finer details of us getting to where we want to.
"For us, the way forward is to try to turn the football we're playing now into something meaningful."
Arsenal manager Mikel Arteta paid tribute to his Arsenal players after they brushed off a string of high-profile absentees to claim an "ugly" win.
This north London derby had been billed as potentially make-or-break for Arsenal after they dropped points at home to Brighton before the international break.
Manchester City's fourth consecutive Premier League win on Saturday coupled with the unavailability of captain Martin Odegaard, Declan Rice and Mikel Merino cast doubt over Arsenal's title credentials, but they answered them superbly in N17.
"We had good moments and others where we had to suffer. We suffered because we had to adapt the plan because of the players that we had available," Arteta said.
"In order to love the game and win you have to do things that people call ugly. Enjoying those ugly things is a big compliment to this team right now."
- with AAP
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