It looked ominous in the first 20 minutes as Celtic struggled to put passes together, trailing to a third minute goal from former Accrington Stanley starlet Josh Windass with Callum Paterson also on the team-sheet of the Yorkshire side.
Liam Shaw had a decent penalty claim overlooked before the 30 minute break but in the second period there was a bit more purpose about the way that Celtic played.
An opportunist strike from Ajeti in the first minute of the second period got Celtic back on level terms, in the 55th minute Moffat put Celtic in front with a strike that he won’t forget, whipping home a half chance from the edge of the penalty box.
Scott Bain for Vasilis Barkas was the only change made on the half hour, for the final 30 minutes the entire outfield was replaced with Leo Heled and Ewan Otoo catching the eye.
In the final minute Odsonne Edouard found the net from a through ball down the right channel as the Postecoglou era got off to a winning start with a few younger players pushing their chances for greater involvement.
CELTIC FIRST XI: Barkas; Ralston, Welsh, Dane Murray, Montgomery, Soro, Shaw, McInroy, Owen Moffat, Henderson, Ajeti
CELTIC SECOND XI: Bain; O’Connor, Dane Murray, Hjelde, Ben McPherson; Luca Connell, Ewan Otoo, Robertson, Barry Coffey; Afolabi, Edouard