Spurs won the second-leg of their round 16 tie 3-1, moving into the quarterfinals 3-2 on aggregate.
Wilson Odobert (2) and James Maddison had Spurs 3-0 ahead before AZ struck after the hour mark through Peer Koopmeiners.
A pleased Postecoglou said afterwards: "Yeah, really liked the way the boys addressed the challenge tonight. It's not easy when you're 1-0 down before the game starts. You know you don't want to give the opposition any opportunity to get a further advantage. I thought we started the game really well.
"Really composed and controlled. We didn't create a lot but at the same time we just had them camped in their half and they didn't really venture anywhere near our goal. Scored a really good goal from our press. And second half started brilliantly, exactly what we wanted to do, got our goal, and then we gave them a goal.
"That puts the tie again on edge a little bit. But I thought our response to that was really good, scored another good goal, probably should have got another one or two to finish the game off.
"We probably didn't handle the last 10 minutes very well. But at the same time when I'm having to change the backline so much, at such a critical time, it's going to disrupt us. Again I thought at the big moments, the boys stood up well. Really pleased, they did a fantastic job to get us through."
On the performances of youngsters Odobert and Lucas Bergvall, Postecoglou was especially happy.
"Yeah, massive, brilliant for us, brilliant for the club, like I keep saying, we've got some really exciting talent. Lucas is growing all the time. People are getting a glimpse of what Wilson's all about. Shame what's happened to him this year, and he's still working his way from fitness. I just thought the whole group handled it really well. It was great to have Romero and Mickey, we just looked so solid at the back. While they were out there, we never really looked like getting threatened. Just made us look more solid as a team. So a lot of positives there."
Beyond his young players, Postecoglou was also eager to highlight the work of Heung-min Son on the night.
He added, "Sonny gives you everything every game, there's no doubt about that. He works so hard for the team. Had a couple of moments today, a couple of times unlucky, a couple of times could have been on the end of things.
"But it was a collective effort tonight. I think it's the first time - somebody said - since October we've had our four leaders on the park: Vicario, Romero, Maddison and Sonny. So much this year we've relied on one or two of those. But to have all four of them out there makes a difference for us. I thought they all led the team well."