Spurs have won just once in seven games in all competitions to pile the pressure on Postecoglou's position.
Cole Palmer scored twice from the penalty spot in the second half to turn the game around, while Jadon Sancho and Enzo Fernandez were also on target for the Blues, who go second in the Premier League.
Tottenham remain down in 11th and defeat was compounded by the loss of influential centre-back Cristian Romero, who had only just returned from injury.
"It is a sore one, a painful one," said Postecoglou. "We started the game really well but as our season has been going, we lose Romero after 20 minutes and that disrupts us a bit and they score straight after."
Dominic Solanke and Dejan Kulusevski had given Tottenham a perfect start inside 11 minutes.
And they were left to rue missing a number of other big chances as Pape Sarr hit the bar, while Solanke and Son Heung-min wasted huge opportunities with the game still in the balance.
Yves Bissouma and Sarr were also guilty of reckless challenges to gift Chelsea two spot-kicks.
"I thought both penalties were poor decisions by us, we have to show more discipline. In the end the critical moments got us," added Postecoglou.
"We had moments to get our third and at half-time we were in a decent position. The second half is a ding-dong, especially at the start, they scored but we had a big chance with Son, we didn't take it and then they go 3-2 up."
'Blame me' says Son
Son did pull a goal back in stoppage time, but the Spurs captain accepted responsibility for the defeat after he missed a one-on-one with Robert Sanchez when the match was delicately poised at 2-2.
"It feels like we lost because of these small details. On such a big stage, you need to step up and score in these moments. I feel like I let the team down and I feel very sorry for the team," said Son.
"I can stay here all day talking about the mistakes but I'd rather blame me with the chance and I'd rather take the blame.
"I had a bit of a different thinking when I was running to the ball. I'm also human and I miss. I feel the pain because it's such an important moment of the game."
A fourth consecutive Premier League win for Chelsea moves them to within four points of leaders Liverpool.
Enzo Maresca, though, reiterated his belief that the inexperienced Blues are not yet ready for a sustained title challenge.
Chelsea's task in north London was made much more difficult by two slips from Marc Cucurella that led to Spurs' first two goals before the Spaniard changed his boots.
"Arsenal, (Manchester) City and Liverpool probably don't slide -- like Cucurella did," joked Maresca.
"To be serious, we are not ready, we are far from these teams but we focus on day-by-day and trying to improve the team."
© 2024 AFP