Pedro Porro and Dominic Solanke struck the goals as Spurs reached the final 5-1 on aggregate.
Afterwards, Postecoglou said: "The players were outstanding. They were well-prepared tonight, the staff did a great job of making sure the boys were really clear about what we needed to do.
"They executed it perfectly. It's been well-chronicled in the last few days it's a difficult place to come through, they've got a really imposing home record, not just with wins but with goals, I thought the lads just handled it really well, as they have throughout this European campaign, particularly the knock-out stages, and credit to them, I thought they did an outstanding job.
"It's a great opportunity. It's my fourth year in European football, plenty of people have been in European football for longer than I have, they haven't got to a final. So you understand it's something that you need to appreciate. And I'll certainly do that. I haven't really thought a lot about it, because obviously the focus is on tonight.
"More importantly, just really pleased for the players and the staff, and particularly our supporters. There were a few hundred here tonight who I'm sure will remember this night. And thousands back home. We've given them some real hope and something to dream about, that we can do something special this year."
Critics will keep trying to tear us down
Spurs will meet Manchester United in the final in Bilbao and Postecoglou is convinced winning the trophy would be transformational for the club.
He also said: "So the question answers itself, it doesn't need me to... but what’s happening now is people are fearing that. That it actually might happen and let's see how we can tear it down somehow and diminish it somehow by saying it's been a poor season and we don't deserve this or we don't deserve that.
"Or somehow comparing us to Man United. Maybe if we had Man United’s success then maybe I’d have a different view. But I came into this role, you know better than me, I could have been sitting here at exactly the same time in fifth position and I can guarantee that the commentary around me 'well that's great Ange but this club needs to win something'.
"That's exactly what everyone would be saying. So of course it’s massive. Of course it is, because you have to frame it against what this club has been through over the last 15 or 20 years and what the supporters have been through.
"Man United have got their own journey to go on. Maybe if they go 15 years without winning something they'll change their perspective on things as well."