Postecoglou lost three of his first six league matches, but Celtic are unbeaten in 18 since.
Reo Hatate was the star of the show on his Old Firm debut as the Japanese midfielder scored twice and made the other goal for Liel Abada.
Victory takes Celtic one point clear of their Glasgow rivals at the top of table.
Postecoglou's men have overturned a six-point deficit at the winter break in the last four games and will now be confident of taking back the title with a number of key players to return from injury and international duty.
"Going top is a sign of the progression. We lost three of our first six games and everyone knows how the Scottish Premiership works, we have had to be nearly perfect since," said Postecoglou.
"It was a big game, we knew the consequences. There was big pressure on us and they handled it".