The Australian coach – sacked by Tottenham Hotspur for their woeful 17th-placed finish in the Premier League, despite delivering the Europa League – replaced Nuno Espirito Santo on Tuesday, giving him little time with the Forest players before Saturday's game.
His teams are known for playing aggressive, sometimes reckless, attacking football, but Forest were flat and barely laid a glove on Arsenal.
Postecoglou, however, said his new charges played creditably after the upheaval of Nuno's sudden departure, something he thought Forest "can put to bed and now start looking ahead".
"It's no secret, it's been a very disruptive week for the players," he told reporters.
"Seven days ago, their world was totally different from what it is today, and you've got to understand that – I certainly do.
"I can't fault the players' endeavour and effort. I thought we hung in there. We never really had control of the game at any point but we still, I thought, worked really hard in the first half.
"It was kind of small seeds towards the end of the first half of us maybe being able to sort of cause them some problems, but the second goal straight after half-time killed the game."
Postecoglou said he had six games in all competitions before next month's international break to "start bedding down some of our principles in terms of how we want to play".
But he was emphatic that he wanted to see changes in Forest's play as soon as their midweek League Cup game against Swansea City.
Asked if it would take weeks or months for Forest to play in his image, Postecoglou replied: "No, mate, it won't be months, it won't be weeks, it'll be Wednesday.
"I can't afford to waste time. This is not a project. This football club (had) a strong season last year, needs to be stronger this year.
"My job is to get us to where I want us as quickly as possible. So by Wednesday night we start seeing some real principles being embedded.
"It won't take long and I'm not going to let it take long."
Check out the match report from the Emirates Stadium with Flashscore.