Forest announced on Tuesday that Postecoglou would take over as replacement for Nuno Espirito Santo, who was axed three games into the new season despite only being given a new contract in June after guiding Forest to the Europa League.
Postecoglou arrived at Forest's training facilities on Tuesday morning local time, and the club announced his appointment little more than an hour later.
Club owner Evangelos Marinakis hailed Postecoglou as a manager with "a proven and consistent record of winning trophies".
"His experience of coaching teams at the highest level, along with his desire to build something special with us at Forest, makes him a fantastic person to help us on our journey and achieve consistently all our ambitions," Marinakis said in a statement.
"After gaining promotion to the Premier League, then building consistently season after season to secure European football, we now must take the right step to compete with the very best and challenge for trophies.
"Ange has the credentials and the track record to do this, and we are excited he is joining us on our ambitious journey."
When Nuno was sacked Postecoglou, who was fired by Tottenham just 16 days after winning the Europa League, quickly emerged as the frontrunner to replace him.
After two years at Spurs, Postecoglou will now have another shot at the Premier League and European football, starting with an early return to North London when Forest face Arsenal at the Emirates on Saturday.
Marinikas's Greek heritage and respect for Postecoglou, 60, appears to have contributed to him hiring the former Socceroos boss for his second EPL job.
In July, Marinakis, who also owns Greek champion Olympiakos, presented Greek-Australian Postecoglou with an award recognising him for being the first Greek to win the Europa League.
"What he achieved, he did with a team that has not won any titles, it has had a very difficult time in recent years," Marinakis said in June.
"In this huge success that the whole world saw, he promoted Greece.
"We must thank him especially for this and we wish him well, although we are sure that he will do well as he has the ability.
"Wherever he goes, the successes will come."
Postecoglou has maintained a dignified silence since being let go by Tottenham, who finished 17th in the EPL last season, despite their Europa success.
Spurs have since started this season brightly under new manager Thomas Frank.
Nuno and Marinakis had an increasingly volatile relationship which - rather than on-field results - led to the split.
Marinakis was seen remonstrating with Nuno after a 2-2 draw against Leicester at the end of last season, although at the time the club put that down to Taiwo Awoniyi playing on with a serious abdominal injury.
Previously, Nuno had criticised Forest's activity in the summer transfer window, saying they had wasted a good chance, despite adding the likes of James McAtee, Omari Hutchinson, Arnaud Kalimuendo and Douglas Luiz.
Having led the club into Europe, Nuno departs the City Ground as the club's most successful manager since Frank Clark in the mid-1990s, and they currently sit 10th in the Premier League, with four points from three matches.
Nuno was Marinakis's seventh manager since becoming Forest's majority shareholder in 2017.
