The 1-0 win, courtesy of Dominic Solanke's penalty, saw Spurs win their quarterfinal overall 2-1 on aggregate.
Postecoglou said afterwards: "Yeah, I'm delighted with many things, super proud of the lads. Going away from home, a quarter-final against a very good side, you know you're going to have to work really hard, be really disciplined and we needed to score. We had to show the quality we had as well.
"To a man I just thought they were outstanding, so proud of them.
"In such a big game, they delivered a performance and it means we're through to a semi-final. I think over the two legs we deserved to go through. It could have been all over in the first leg but we earned the right to be in the semi."
On Tottenham's chances of winning the competition, Postecoglou insists they can't look that far ahead.
He continued: "No, I don’t think that way. From our perspective we’ve had a difficult season and I think there’s some reasoning behind that was pretty evident that when we've got our strongest team out there, a lot of things we’ve suffered with this year I’m sure we wouldn’t have.
"The spine of our team, at times we've been missing Vicario, Romero, Van de Ven, Bentancur, Solanke, all of them we've been missing at different times this year.
"We've got the whole group. It's not just them, it's the whole group together. We know that we can be a team that's pretty compelling against anybody. We're in the final four now so we've given ourselves a chance."
"You're going to have to put up with me for longer"
Many claimed last night's result would decide Postecoglou's future, though he insists there was no sense of relief after the victory.
He also said, "I think I said yesterday, I am the same manager today that I was yesterday so if people think us winning tonight makes me a better manager or whoever thinks I wasn’t doing a good job yesterday, should be feeling the same way.
"I don’t care, it doesn’t bother me, it doesn’t affect what I do. For me, it’s always about the dressing room. Do the players believe? Do the staff believe? That’s much more important than what others may make of me.
"So, unfortunately for a lot of you, you’re going to have to put up with me for a little bit longer, mate, so let’s see how that goes."