Spurs have made an impressive start to life without Kane following his €120 million ($127.8m) move to Bayern Munich by taking 13 points out of a possible 15 in their opening five Premier League games.
However, Kane's absence may be more keenly felt this weekend given the 30-year-old is the all-time top goalscorer in derby matches between Arsenal and Tottenham, netting 14 goals in 19 games.
"There was a risk of feeling that void right from the start," Postecoglou said on Friday. "That's the reality of it. It is a fairly significant figure on and off the field to be fair that left very late in the piece. But we kind of tried to, as a collective, focus on the road ahead, the kind of team we want to be.
"I've been really pleased with the way the lads have tackled the task of us being a real attacking threat from all areas of our game and so that people don't see a glaring gap there. But you are still trying to replace arguably the greatest player this club has ever had who still was making a very, very significant contribution. The opportunity for the void to be there is there all the time but we've managed that well so far."
Pushed on whether he was confident Spurs had the personalities to step up without Kane, Postecoglou said: "We'll learn that. We're five games in. I understand that for many people, they will see this as our first real test and I get that. It is pretty easy, you could almost write two stories now: if we are successful, great we're on the right track. If we're not, we've still got a long way to go.
"For me what is going to be more important is how much of our football I see in a big game like this. How much of us can I see against a top opponent? That will give me the biggest indicator of where we're at but ultimately we'll still be six games into a new cycle, a new group of players, a very young group of players.
"Either way, irrespective of the outcome, I'd say the performance will be my biggest measure but knowing we have still got a long way to go."
Spurs have not won a league game at Arsenal since 2010, losing on eight of their last 12 visits but Postecoglou said the fixture's history should have no bearing on Sunday's match.
"Ultimately the challenge is they are a very good football team and I can't get away from that," Postecoglou said.
"They are a very good football team, so irrespective of past records. If we had beaten them twice last year, I would be going into this game saying it will be a tough game and it will be a tough game for our players, particularly away from home.
"That is the challenge for us. I have never put too much stock on previous records, previous history because I certainly wasn't involved in any of those and some of the players weren't involved, so there is no point us reflecting on the historical context of the fixture.
"What is really more relevant is they are in good form, we're in good form and we want to go out there and try to play our football against them."
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