Neco Williams had Forest ahead on two minutes before they were pegged back on 20 minutes by Jaidon Anthony.
Postecoglou said of Williams' goal: "Great strike. He's been unlucky with one in the second half as well. We want our full-backs to join in with those sort of situations as well. We still have areas of improvement but we responded well in the second half.
"We created a lot of chances. It wasn't just individuals, it was combinations and if we keep doing that then eventually a goalkeeper will throw one in for us.
"It's challenging but that's what we do. I've seen progress in the last seven days. I know it was Arsenal away but the general approach, we've made enormous progress. All that takes time but that's the exciting bit."
The Austalian also said: "It's a difficult place to come. Burnley have a fantastic home record and the way they play makes it a challenge. I'm really happy with the way we played in certain parts of the game. They're always a threat with counter attacks and transitions."