Postecoglou, who was sacked as Tottenham Hotspur boss in June, 16 days after leading them to Europa League glory, started his new role with defeat to Arsenal before losing to second-tier Swansea City in the League Cup in midweek.
Burnley forward Jaidon Anthony's equalizer to earn the hosts a 1-1 draw on Saturday left Postecoglou winless in his last nine Premier League games, his longest run without a league win as a top-flight manager across spells in Australia, Japan, Scotland and England.
"It is always a tough game here," Postecoglou told the BBC. "I am really happy with certain parts, we controlled it well and their keeper pulled off some great saves.
"We lost a certain amount of control but disappointed we didn't take all three points. We had a couple of sloppy moments and it allowed them to get a bit of traction in the game. We still created enough chances to win the game.
"From seven days ago from when we played Arsenal, we look a different team and we saw elements of that on Wednesday night too. I am disappointed but, at the same time, the progress in our play is the most positive thing for me."
It is back to the Australian's favourite cup competition next as he looks to secure a first victory when Forest face Real Betis in the Europa League, a competition which holds fond memories for Postecoglou, even if it was not enough to earn him an extended stay at Spurs.
"I'm really looking forward to it," he said. "It'll be great for this group of players.
"I have three years of experience in the Europa League. Last year went well but it'll be great for these players."