The Australian, who replaced Nuno Espirito-Santo at the City Ground this month, looked stunned as Socceroos defender Cam Burgess fired in a rebound deep in added time to send the home fans wild on Wednesday.
"We have a team here that works so hard and anything can happen if you go to the final whistle," said match-winner Burgess.
"It was a bit test for us to go up against a Premier League team and show what we can do. If that's a sign of things to come then it could be a great season."
Forest, four-time winners of the competition, had controlled the opening half with Brazilian Igor Jesus scoring twice - his first in the 14th minute when he fired in at the far post after a ball rebounded into his path and the second just before the break after a flowing move.
There was no hint of the drama to come for most of the second half on a rainy south Wales night but Swansea got a lifeline with just over 20 minutes remaining as Burgess found space to head in from a corner.
Slovenian substitute Zan Vipotnik then levelled in the third minute of added time and Forest could not even get to a penalty shootout as Burgess, who joined from Ipswich Town in the summer, pounced after Ethan Galbraith's piledriver came back off the woodwork.
It was acting skipper Burgess' first brace of his career, the Swans remain unbeaten on home turf since mid-March and the 2013 winners advanced to the last 16.
"I just thought it was a game we really should have killed off long before it got to the stage that it did," Postecoglou, the former Tottenham Hotspur manager, said.
"You have to know the warning signs in cup football. If you let sides in, there is always a chance they could do what they did to us.
"I thought we scored a good third goal, which was disallowed, and the guys got a bit comfortable in that moment.
"A couple of decisions didn't go our way and ... we've paid a heavy price ... They had two shots from outside the box in the first half and they didn't really test us until the end."
Postecoglou's side, who face Burnley in the Premier League in a bid to return to winning ways, have now conceded six goals in his two games after his Forest reign began with a 3-0 defeat at Arsenal on Saturday.
With PA.
