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.
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.
"I got the sense we were getting a bit comfortable and thought the game would finish off easily.
"But that's not what happens, particularly in cup football."
Postecoglou's side have now conceded six goals in his two games after his Forest reign began with a 3-0 defeat at Arsenal in the Premier League on Saturday.
With PA.
