Brennan Johnson's well-taken opener marked his sixth goal in as many games across all competitions before James Maddison's first-half finish had Spurs leading 2-0 at the break on the south coast.
It was just the second time that Tottenham lost a Premier League game they led by two goals in their last 166 such matches (W159 D5), and Postecoglou fumed at the full-time whistle.
The Tottenham manager said on Sky Sports: "Disappointing, frustrated and absolutely gutted with that. It's the worst defeat since I've been here.
"Unacceptable second half, and nowhere near where we should be. We got carried away with how we were going.
"We kind of accepted our fate [in the second half] and it is hard to understand as we've not done that while I've been here. We paid the price.
"The problem is we are travelling along too smoothly, football and life will trip you up if you get too far ahead of yourself."
Sunday also marked the 10th time that Tottenham have lost a Premier League game in which they led by two or more goals, becoming the first club to hit double figures for this unwanted record.
"There is no message," the Australian said when asked what he would tell his players. "It is a terrible loss for us, as bad as it gets. There is only one way to fix it and that's my responsibility.
"We lost all our duels and if you're not competitive it is not going to work; we were not competitive.
"They all go on international duty the majority of them, they will process it individually and I will process it when everyone gets back."