First-half goals from Elliot Anderson and Chris Wood were enough to get Nuno Espirito Santo's side back to winning ways, despite the hosts dominating for large periods at Tottenham Hotspur Stadium.
Anderson's deflected strike opened the scoring in the fifth minute after Spurs failed to deal with a corner, while Wood headed his 19th league goal of the season to double Forest's lead in the 16th minute.
They eventually made their dominance count with Richarlison's header halving the deficit three minutes from time, but it proved too late as they suffered a fourth defeat in five league games.
Postecoglou's team has collected just 18 points since beating Manchester City 4-0 on 24 November (AEDT), with only the bottom three sides recording fewer during that time.
"We conceded two poor goals and gave ourselves a mountain to climb. I still think we didn't deserve to lose on the balance of play.
"Our football was good, there's no denying that. We created loads of opportunities, we had them pegged in. But like I said, it was really disappointing to concede the two goals like we did.
"It's hurt us all year, we've let ourselves down. It's a game we should have won tonight. There shouldn't have been a slow start.
"That's the frustrating bit. It's not the only time it's happened this year. We have to accept the fact we haven't been at the level we should be.
"I don't like losing, we've lost too many games. We've got to keep working and try and rectify these things, so the football we play gets the results it deserves."
Spurs were fresh from UEFA Europa League heroics last week, when they booked a place in the semi-finals after securing a hard-earned 2-1 aggregate victory over Eintracht Frankfurt.
Kulusevski acknowledged fatigue may have been a factor against Forest, but insisted it was no excuse for Spurs being slow out of the blocks.
"[It was a] sloppy start," he said. "We conceded three goals in 10 minutes. Yes, one was offside, but then after that, it is difficult to get back in.
"We also played on Thursday night [Friday morning AEST], but you have got to be ready from the start. You come back from a long trip at five in the morning, but if you are not on it then you will pay. That's what we did.