Forest had looked in complete control after Igor Jesus struck twice in the first half, putting the Premier League side on course for a place in the last 16.
But Cameron Burgess pulled one back after the break before a chaotic finale saw Zan Vipotnik equalise and Burgess grab his second to complete a stunning turnaround for the Championship club.
“We had to finish them off and we had plenty of opportunities to do that,” Postecoglou said. “I think the lads got a bit comfortable in that moment and you’ve got to know the warning signs.
“Players just thought it would naturally finish that way and we’ve paid a heavy price for that. We need to improve.”
It was the second successive game Forest conceded three goals under Postecoglou, following their 3-3 Premier League draw with Crystal Palace last weekend.
The Australian, who replaced Nuno Espirito Santo in the summer, admitted the defeat was a huge setback in his efforts to stamp his style of play on the side.
“Tonight was a great opportunity for us to progress in a cup and give ourselves a chance of success,” he added. “To let it slip is hugely disappointing. There were positives in individual performances, but ultimately the overriding factor is that we are out of the competition when we really should have gone through.”
Postecoglou, who won the Europa League with Tottenham last season before being sacked, insisted the focus must now turn to Saturday’s Premier League clash away at Burnley.
“It’s not about me, it’s about the club,” he said. “We have to create an environment where we can have real success. That’s the job in front of us.”
