The Hoops lost 2-1 in an entertaining tussle at Tynecastle, John Souttar, the brother of Australia defender Harry Souttar, heading home the winner in the 89th minute.
Celtic fell behind early after a defensive mix-up before Anthony Ralston equalised with a fine solo effort, but Postecoglou admitted his side should have done more with an astonishing 74 per cent of possession across the 90 minutes.
"Hugely disappointing. After a chaotic first five or 10 minutes I thought we grew into the game, we kind of controlled the game.
"Some of our play in the front third wasn't great – we didn't really capitalise on the possession we had – but we got ourselves back into the game and thought at that point we looked like we'd go on with it, so hugely disappointing to concede one at the end."
Celtic faces another European trip next Friday (AEST), when it will meet Czech side Jablonec in a UEFA Europa League qualifier, the Hoops having missed out of UEFA Champions League qualification last week after a 3-2 aggregate loss to FC Midtjylland.
Postecoglou has two home games coming next in the Scottish Premiership against Dundee and St Mirren before facing the champion Rangers in what will no doubt be a keenly watched opening Old Firm Derby on 29 August at Ibrox.