Cameron Carter-Vickers bagged a fortunate goal on debut, scoring the opener with a deflected looped shot after Postecoglou was forced to make five changes to his starting line-up because of departures and an injury to Kyogo Furuhashi.
Fellow Australia Tom Rogic got a start and Swiss striker Albian Ajeti more than covered for the loss of the Japanese with two goals in the match.
The teams had gone in level at 0-0 at the break, but whatever Postecoglou said in the sheds was left ringing in the ears as the Hoops took full control in the second period having been frustrated in the first.
The 64th minute brought the opening goal for Carter-Vickers, and Ajeti scored in the 70th and 85th minutes to put the game to bed.
“It was testing. When you don’t score early it always has the potential to turn into a little bit of a struggle, but for the most part, we maintained our discipline and sort of knew in the second half it would open up a bit," Postecoglou said.
“We got some opportunities and once the first goal went it, the game opened up a little.
“It wasn’t easy, we didn’t expect it to be easy, but credit to the players, they stuck at it.”
Celtic sits fifth on the Scottish Premiership table after five matches, with arch rival Rangers back on top of the pile after come-from-behind 2-1 win at St Johnstone.
Hibernian can reclaim top spot on Sunday (AEST) with a win, but so too can its opponent, Hearts, with both sides sitting on 10 points behind Rangers, which has 12.