Griffiths penned a new one-year contract earlier this week after much speculation about the 30-year-old’s future at the club.
The Scotland internationalist has scored 123 in 261 games for Celtic and Postecoglou believes that the player has “unfinished business” in Glasgow.
“I try not to have pre-conceived ideas about anyone,” said Postecoglou
“Whenever I've gone into a new position I'll always back myself to make my own assessments and with Leigh I just saw a footballer who I don't think is old in football terms by any stretch.
“He's a goalscorer, which we all know is the most difficult position on the field to find players for.
“He knows Celtic, he knows the league and I had a good chat to him.
“I think even himself he wants to make sure his legacy at this football club is a positive one right up until the end, he probably feels like he has some unfinished business.
“I'm more than happy to give him that opportunity, and with the kind of football we play I believe he can score goals.
“I'm happy to add him to the squad.”
Postecoglou has also reinforced his squad with the signing of Sheffield Wednesday defender Osaze Urghoghide.
The 20-year-old Dutchman can play across the back four and Postecoglou revealed that Urghoghide “responded well” to conversations about his potential career at Celtic.
“The club had done some scouting work and there was a whole range of players they were looking at before I came in,” said Postecoglou.
“I had a look at all the players who had gone on the list of potentially bringing in and he [Urghoghide] was one I was really keen to talk to.
“It was a big decision in his career, obviously he wouldn't know me too well but he certainly knows Celtic as a football club and what the staff have done for young players that have come through here.
“I just spoke to him about my vision for the football we're going to play here and how he would fit into that."
Postecoglou added on Celtic TV: “He responded really well, it was a big decision for him because he had some big options ahead of him but he chose a development pathway and that showed me he's the right type of character we need here and I look forward to working with him.”
Why Urghoghide was signed
Urghoghide and Griffiths are expected to be involved when Celtc play the first of their pre-season friendlies next week.
Postecoglou takes his squad to Wales, where they will play three matches in the space of seven days in Newport.
On Wednesday, July 7, Celtic take on Sheffield Wednesday at Dragons Park (kick-off 3pm) and then on Saturday, July 10, they face Charlton at the same stadium at 3pm.
Celtic conclude their action in the south-west of Britain with a game against Bristol City on Wedneday, July 14 at the Robins’ training ground.
All the matches are expected to be played without supporters present.