Mark O'Hara headed the Paisley side ahead in the first half, before Jonah Ayunga added a second after the break to inflict a first defeat on Celtic since September 19 last year.
The loss was just Celtic's second in their past 27 visits to St Mirren, and Postecoglou accepted his side was well off its usual standards.
"We just weren't aggressive both in action and in thought," he said. "We kind of tried to ease our way into a game of football and you just can't do that. You've got to be at it from the start and we weren't.
"You don't put it down to one of those days; it's a disappointing day. We pride ourselves on having certain standards and we didn't reach those. And football will always teach you that lesson.
"What has got us to this point over the past 15 or 16 months is having a certain work ethic and certain levels of performance and belief. We didn't reach any of those. We've got to take the hits when they come and we've just got to get back up and go again.
"We've done that before as a team over the past 15 or 16 months. It's a sore one, it's one that we've got to use as the fuel we need next to get going again.
"We were nowhere near the levels we want to be. Credit to St Mirren. They worked really hard and were really committed to their game plan. They made it hard for us, but we never hit the levels we need to play our football. It's about dusting ourselves off and getting going again."
Celtic has a two-point lead over bitter rival Rangers at the Scottish Premiership summit ahead of the international break.