Richarlison revealed this week he has had some difficulties off the pitch, but he came off the bench to produce a stoppage-time goal and assist.
"We’ve got to maintain a balance in life and even for him, he needs to understand his football is still good, he’s a fantastic footballer," said Postecoglou.
"He’s got so much to give and that can help ease the burden he’s feeling in other parts of his life, like so many of us.
"I’m really pleased for him – I thought he did really well. We were obviously chasing a goal and getting some balls in the box and he’s always a threat in the air."
Postecoglou was in danger of suffering his first defeat just one day after winning the Barclays Manager of the Month for August, and he was delighted that his side battled all the way to the finish.
"It’s a great ending to the game," he said. "In terms of what we’re trying to build, there’s the football side but also it’s about building the spirit and character and resilience within the group and they have showed bags of it today.
"Every time you have to deal with adversity and you overcome it, I think it helps in what you’re trying to grow.
"It would have been easy for our supporters to get frustrated, but I do think our supporters are seeing in this team, even when things weren’t going to script today and we were 1-0 down, they can see the team trying to play the kind of football they want to see and I think that helps. They can resonate with that."