Spurs finished 17th after a second-half turnaround by the visitors more than cancelled out Dominic Solanke's early penalty, but it failed to spoil a party atmosphere at Tottenham Hotspur Stadium, where the club's first trophy in 17 years was paraded after full-time.
Postecoglou was serenaded before, during and after, and yet there is no clarity over whether he will be afforded a third season in charge despite clinching the club's first European title in 41 years.
"I will be honest, I have been finding it really weird talking about my future when we have done something unprecedented," the Australian head coach said.
"I have had to answer the questions because no-one else at the club is in the position to do so, I guess.
"I have got no doubt, though, that this could be a real defining moment for this club, because wherever I have been, I have made an impact where I have brought success to a club that hasn't had it for a while.
"You just have to look at those clubs' trajectory even after I left, they are still competing for things. I really think this is a moment in time where this club could push on and be a real contender for honours on a yearly basis."
Chairman Daniel Levy's end-of-message programme notes published on Sunday afternoon failed to address Postecoglou's future.
Even though the Australian believes he can take Tottenham to the next level after they finally ended their trophy hoodoo, the 59-year-old hinted at weariness after an emotional week with success in Bilbao followed by Friday's open-top bus parade in front of an estimated 200,000 fans.
Postecoglou added: "You know my gut feeling? My gut feeling is I feel right now that I've done something that no-one believed I could. And I shouldn't be sitting here talking about it (my future). That's my feeling, but it is what it is.
"Probably I am talking now because I'm tired, mate.
"No, you're right to ask the question but you're asking the person who can't give you that answer. And I guess, even for you guys, you wouldn't be asking it if there wasn't a doubt, right? But I can't answer it.
"There is nothing I can say that will answer that question. Other people can, so from my perspective, it doesn't diminish the achievement.
A Jack Hinshelwood double followed by late efforts from Matt O'Riley (penalty) and Diego Gomez helped Brighton secure eighth spot, but results elsewhere meant it failed to earn European qualification for the south-coast club.
