Richarlison, who has scored three goals and registered four assists in 37 appearances in all competitions for Spurs since joining from Everton in July last year, is yet to find the net this season.
The forward was signed for around 50 million pounds ($60 million) plus another 10 million pounds in add-ons, according to British media.
"I am sure Richy wants to score goals, but far more important for me is how he is contributing to our team set-up, and I thought he was good last weekend," Postecoglou told reporters ahead of Saturday's game against Bournemouth.
"From my perspective that is the key thing, and I am sure the goals will come, but there is no point getting the goals if the team is not performing."
Harry Kane, the club's talisman and record scorer, was sold to German giants Bayern Munich on the eve of the new season. Richarlison is expected to take up the mantle at Spurs following the England captain's exit.
"I don't know that there's that much of a burden there to carry. Ultimately, this football club needed to change," Postecoglou said.
"I think everyone at the club's got to grips that Harry is gone. He's gone off to a new challenge, and I'm sure he'll do very well over there. We've got our own challenge now.
"I don't think there's any use in comparing yourself to something that's no longer here and is no longer relevant. It would be different if it was on the back of a successful era."
Postecoglou has only been in charge of Spurs for two top-flight games, but the Australian has already managed to lift the gloom that was hanging over the north London club when he arrived from Scottish Premiership side Celtic.
He has guided Spurs to a 2-0 home win over Manchester United, following up from a 2-2 draw at Brentford, and is hoping to improve on last season's eighth-place finish.
