"I know the significance of it and it's been a hell of a ride so far," Postecoglou said.
"I'm just so proud of this group of players, they've been the most resilient group I've ever come across with what's been thrown at us the last five or six months and I'm super pleased they get their reward.
"We're still at the beginning and we want to enjoy this because this group of players has been through an awful lot and yet risen to this challenge -- including today. We just want to keep going."
Postecoglou left Japanese side Yokohama F. Marinos to replace Neil Lennon at Celtic in June. He has also won the A-League with Brisbane Roar, and managed Australia from 2013 to 2017, where he won the 2015 Asian Cup.
The two sides were deadlocked at half time, despite Celtic almost taking an early lead after defender Greg Taylor's strike was spilled by Hibernian goalkeeper Matt Macey, but winger Liel Abada could not convert as opposing defender Paul Hanlon hooked the ball off the line.
Hanlon was at the heart of the action again early in the second half as he opened the scoring with Hibernian's first shot on goal, heading in from from a corner. But Celtic fired back just a minute later, with striker Furuhashi speeding past Hanlon before slotting home an equaliser.
Furuhashi proved dangerous again on 72 minutes, once more pouncing on a ball over the top of the Hibernian defence before netting the winner with a well-taken chip over the head of Macey.
Information from Reuters contributed to this report.