The Bhoys are on their way to winning the Scottish Premiership title after beating Rangers in the Glasgow derby on Sunday.
With a six-point lead at the top of the league table and six games left in the season, Celtic are not expected to allow Rangers back into the league title race in the coming weeks.
Postecoglou has earned rave reviews for the way he turned Celtic around following their disappointment last season and Grant believes a lot of it has to do with the way he has handled the pressure of being the manager at Parkhead.
He stressed that the Celtic team are a reflection of the Australian given the way they remain calm and calculated even under pressure.
The Celtic legend said on the Go Radio Football Show: “I think the way he has handled himself, especially in Glasgow. You know what it’s like on both sides.
“They are after you, the press are after you if you make one mistake and say something and he always does it with calmness.
“You can see that he is always prepared and is always thinking and I think that is the way his team look.
“He is very clear in what he says and the way he structures his team and the way he structures his press conferences.
“He knows what he is going to say even before he says it.
“He is a very clever man and he is very astute and I think Celtic needed that sort of personality after the disappointment of last year.”
Celtic are on course to complete a domestic treble this season under Postecoglou if they continue on their current form.
