Despite losing star striker Kyogo Furuhashi only five minutes into the first half with a shoulder injury, the Hoops opened the scoring just three minutes later when Liel Abada's powerful strike beat the outstretched hand of Jon McLaughlin.
Jota doubled Celtic's lead with a cute chip from a tight angle just after the half-hour mark, before Abada netted his second and the host's third with a fizzed effort through the legs of McLaughlin five minutes before the break.
While the likes of Alfredo Morelos, Scott Wright and Scott Arfield were unable to make an impact off the bench for Rangers in the second half, Celtic substitute David Turnbull added a fourth for Celtic just six minutes after entering the field of play, with a mistake from McLaughlin allowing the midfielder to seal a comfortable victory at Parkhead.
Celtic have maintained their 100% start to the new Scottish Premiership season, with the reigning champions sitting at the top of the table after scoring 25 goals and conceding only one from their opening six matches.
Speaking after the match, Postecoglou emphasised the importance of claiming maximum points in the Old Firm derby in front of their own supporters, and how he has been particularly pleased with his side's "outstanding" form so far this term.
"It was a big game, we knew that. The boys were outstanding from start to finish," the Australian told BBC Sportsound.
"We know what this game means to our supporters. We've given them another memorable day and the players deserve all the credit for that."
When questioned on his views about sitting five points clear of Rangers after just six matches, Postecoglu added: "What's more important to me is how we're playing, that's a better indicator.
"You can win games in many ways, but our form has been outstanding and we've been getting better every week. If we keep going like that, we're going to be hard to stop."
Postecoglou also provided an update on injured duo Furuhashi and Carl Starfelt, adding: "It was a shoulder injury [for Furuhashi]. Carl had a niggle in his knee.
"It's a bit of blow because Kyogo has been in brilliant form so far this season and it was disappointing for him as I know he was looking forward to the game."
As for Rangers, manager Giovanni van Bronckhorst stated that he and his side now need to reflect on their "very disappointing" performance at Parkhead.
"You know the start you have is very important in how the game will develop. We knew the threat they had with quick throw-ins and free-kicks," the Dutchman told BBC Sportsound.
"We were exposed twice, which gave two goals away. For me, that's very disappointing. We knew they were going to do [restart quick] and still we weren't prepared enough.
"The longer the nil is on the scoreboard, the more you can grow in the game. Today, that wasn't the case. I'm disappointed we couldn't give the game we wanted for our fans.
"Everyone needs to reflect in what they did well and what they did wrong, same for me."
Asked about the five-point gap to Celtic at the summit, Van Bronckhorst added: "There are still a lot of games to be played, plus three games against Celtic, so we have to move on and get stronger and make sure the moments we had today can never happen again, no matter who we play."
Both Celtic and Rangers will now turn their attention to two huge Champions League group-stage matches next week, with the Hoops welcoming holders Real Madrid to Parkhead while the Gers visit Ajax in Amsterdam on Wednesday.