Postecoglou was sacked after just eight games and 40 days in charge at the City Ground, making his the shortest ever permanent managerial stint in Premier League history.
Forest picked up just one point from the Australian’s five league games and were also dumped out of the League Cup by second-tier Swansea.
Club owner Evangelos Marinakis left his seat during the match as Chelsea took control in the second half and Postecoglou’s fate was confirmed shortly after full-time.
Forest are already looking for their third manager of the season after Nuno Espirito Santo was dismissed last month.
Chelsea had to survive an early onslaught before two goals in three minutes early in the second period swung the game in the Blues’ favour.
Teenager Josh Acheampong headed in his first senior goal from Pedro Neto’s corner.
Neto curled in a free-kick to double Chelsea’s lead before Reece James added a third late on to lift the visitors up to sixth in the table.
Defeat saw Forest fall into the relegation zone.
Meanwhile, Liverpool slumped to a fourth consecutive defeat for the first time in 11 years as Manchester United won 2-1 yesterday (Oct 19) to end their near decade-long wait for victory at Anfield.
Harry Maguire’s 84th minute header secured back-to-back Premier League wins for the first time in Ruben Amorim’s tenure as United boss.
Twelve minutes from time, Cody Gakpo had cancelled out Bryan Mbeumo’s opener after just 61 seconds.
Defeat leaves Liverpool four points adrift of Arsenal at the top of the table and Arne Slot still searching for answers in how to get the right blend after splashing out nearly £450 million (B19.8 billion) on new players in the transfer market.
Unstoppable Haaland
Elsewhere, Erling Haaland’s unstoppable form is dragging Manchester City into title contention as the Norwegian took his tally for the season to 23 goals in 13 games for club and country with a brace in the 2-0 win over Everton.
After two defeats in their opening three Premier League matches, Pep Guardiola’s men are now unbeaten in eight games in all competitions.
Haaland proved the match-winner once more with two clinical finishes in five minutes either side of the hour mark at the Etihad.
He used all of his 1.95-metre frame to power in a header from Nico O’Riley’s cross to break the deadlock.
Haaland then had a helping hand from Jordan Pickford for his second as the England goalkeeper failed to keep out a low shot to his left.
Leandro Trossard’s second-half goal was enough for Arsenal to overcome a stubborn Fulham as the Gunners won 1-0 at Craven Cottage to stay top of the table, while North London rivals Tottenham fell to a 2-1 home defeat to Astton Villa.
Bournemouth were denied top spot earlier on Saturday by Jean-Philippe Mateta’s 97th minute penalty to complete his hat-trick and salvage a 3-3 draw for Crystal Palace.
Two goals from Eli Junior Kroupi put the Cherries 2-0 up at Selhurst Park at half-time.
Palace lost for the first time in 20 matches in their last outing at Everton.
The Eagles avoided another defeat thanks to the in-form Mateta, who was fresh from scoring his first international goal for France.
Mateta struck twice to level before Ryan Christie turned in Marcus Tavernier’s cross in the 89th minute to restore Bournemouth’s lead.
However, Andoni Iraola’s men were denied another huge win when Mateta stroked home from the spot after wantaway Palace captain Marc Guehi was fouled inside the box.
Newcastle’s struggles on the road continued with a 2-1 defeat at Brighton as Danny Welbeck struck twice.
The Magpies have yet to win away in the Premier League this season and slipped to 11th, behind local rivals Sunderland.
The Black Cats climbed up to seventh with a 2-0 victory over Wolves, which left the visitors five points off safety at the bottom of the table.
Burnley climbed out of the bottom three thanks to goals from Lesley Ugochukwu and Loum Tchaouna to beat Leeds 2-0.