Ousmane Diao tapped into an empty net for the opener but the home fans erupted when Dan Ndoye knocked in the equaliser only for Midtjylland to retake the lead through Mads Bech.
Chris Wood had two goals disallowed for offside and the hosts were caught on the counter by Valdemar Andreasen, who ultimately sealed the win with a third for the Danish side.
Wood tucked away a stoppage-time penalty to halve the deficit but it was not enough to stop boos from ringing around the City Ground as fans chanted "sacked in the morning" in the direction of Postecogclou.
Asked if he was surprised how early the fans had turned on him, Postecoglou said: "The fans are disappointed. They're allowed to have their opinion on it and I heard their opinion.
"Nothing surprises me anymore in football. I guess it's just the climate we're in and it seems that's the way things are going.
"But that's nothing I can control, my responsibility to make sure we try and get some wins for this football club and progress it and that's what I'll be focusing on."
The result left Postecogclou still without a win since taking over - losing four from six - and became the first permanent Forest boss to be winless in his first six games in 100 years.
The Australian has faith he can turn things around though.
He added: "Yes I do (believe things can turn) and nothing that I saw tonight changes any of that.
"I get the mood around the place isn't going to be great. I understand people's kind of attitude at the moment, particularly towards me, but I never worry about that, it's not unfamiliar territory for me.
"I really believe that we're on the right track and when we get through this period of uncertainty we'll be on a good path.
"Of course I prefer people were optimistic in what I'm doing, I get that, but I can only change that by winning games of football. I really believe the process in what we're doing and don't think we're far from getting the results we need."
Midtjylland earned a second straight win in the competition and sit second in their standings following their victory.
Boss Mike Tullberg was proud to pick up a win over a Premier League side in what he describes an "amazing" atmosphere.
"We played an incredible team from England and the guys did very well," said Tullberg.
"The home crowd was fantastic at the start of the game. To be in this atmosphere was amazing. Really, glad for us, a great win and hope Forest get their goals for the season."
