The defeat left the Australian under mounting pressure, with Forest owner Evangelos Marinakis reportedly storming out of his seat before full-time while trailing 2-0.
Postecoglou's side began brightly, creating several early chances but again lacked composure in front of goal. Elliot Anderson hesitated when clean through after a Chelsea error from Trevoh Chalobah, while Morgan Gibbs-White volleyed acrobatically over from a tight angle following a high press turnover, one of multiple let-offs for the visitors.
Chelsea, without key midfielders Cole Palmer, Moises Caicedo and Enzo Fernandez, looked flat for much of the first half. Their only real opportunity came when Joao Pedro's clever dinked pass sent Andrey Santos through, but the Brazilian shot just wide.
Enzo Maresca, who watched from the stands due to a suspension, made bold half-time changes that transformed his side, bringing on Caicedo, Jamie Gittens and Marc Guiu.
Pedro Neto sparked Chelsea into life, beating his marker before delivering a perfect cross for 18-year-old Josh Acheampong to head home his first Premier League goal in the 49th minute.
Just three minutes later, Neto doubled the lead himself with a superb free-kick from the edge of the box that left goalkeeper Matz Sels rooted to his line.
Forest's frustration deepened when Igor Jesus hit both the bar and post in quick succession from Callum Hudson-Odoi's inviting cross in the 70th minute, summing up their dismal afternoon.
Reece James sealed the win in the 84th minute, thumping in from a half-cleared corner after Sels weakly punched clear, capping an impressive away display, Chelsea's fourth victory in their last five visits to Forest.
Malo Gusto's late red card in the 87th minute for a second yellow took some gloss off the result, but Maresca's men showed renewed confidence.
Postecoglou, meanwhile, cut a forlorn figure at full-time, declining immediate comment amid growing speculation over his future. With Forest still searching for their first league win under his leadership, his position looks increasingly precarious as the club weighs its options.
Roberto De Zerbi wants Tottenham Hotspur to turn the fighting spirit they showed against Manchester United into a stronger team as they prepare for their next Premier League match.
Morgan Rogers has credited Xabi Alonso's team talks for Chelsea's impressive starts to matches this season after the Blues raced to a 5-1 victory over Bournemouth at Stamford Bridge.
Manuel Akanji has admitted he is unsure what will happen to Manchester City's trophies following the club's guilty verdict over financial rule breaches, but the Inter Milan defender hopes he will get to keep his titles.
Michael Carrick has praised Mikel Arteta for building a successful team at Arsenal but admitted that the Spaniard's achievements are difficult to celebrate because of Manchester United's historic rivalry with the Gunners.
