The Australian, who replaced Nuno Espirito-Santo at the City Ground this month, looked stunned as Cameron Burgess fired in a rebound deep in stoppage time to send the home fans wild.
Forest had controlled the opening half with 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 stoppage time, and Forest could not even get to a penalty shoot-out as Burgess pounced after Ethan Galbraith's piledriver came back off the woodwork.
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.