The pre-season friendly - played behind closed doors across two 30-minute halves - offered Hoops fans an early glimpse of life under the Australian manager.
In a match that pitted the former Socceroos coach against one of the key players in his Australia side's run to the 2015 Asian Cup title - Massimo Luongo - it was Wednesday which started brighter, pressing Celtic high and grabbing the opener through former Rangers attacker Josh Windass.
With both sides fielding different line-ups in each half, Celtic gradually settled into the match, increasingly pushing Wednesday back with its high-tempo possession game.
If the first of three 30-minute periods belonged to Wednesday, the second was all Celtic, as Albian Ajeti drew the hoops level right after the first break.
Owen Moffat fired Postecoglou's men in front towards the end of the second period, before Odsonne Édouard rounded off a comfortable first hit-out under the new manager in the 89th-minute to make it 3-1.
“We grew into the game," Postecoglou said. "We played some decent stuff, scored a couple of good goals and finished strong.”
CELTIC FIRST XI: Barkas; Ralston, Welsh, Murray, Montgomery, Soro, Shaw, McInroy, Moffat, Henderson, Ajeti
CELTIC SECOND XI: Bain; O’Connor, Murray, Hjelde, McPherson, Connell, Otoo, Robertson, Afolabi, Coffey, Edouard
SHEFFIELD WEDNESDAY: Wildsmith; Borner, Dunkley, Hutchinson, Palmer, Trialist, Bannan, Luongo, Green, Windass, Paterson Subs: Dawson, Galvin, Brennan, Dawodu, Waldock, Hunt, Hagan, Dele-Bashiru, Adedoyin, Trialist