Jacob Ramsey gave Villa a first minute lead, with Morgan Rogers also scoring for the hosts. Mathys Tel struck late for a Spurs consolation.
Postecoglou said afterwards: "Obviously disappointed we don't progress. It wasn't a great start, that's the last thing you want, and we really struggled to get some control. Then we had an enormous chance at 1-0 we didn't take to equalise.
"We got a better grip of the game but unfortunately fell short. When you go a goal down it gives the opposition momentum and confidence. You want to wrestle control early and we didn't do that.
"We were still in the game (at halftime). We just needed to hang in there. The second half was a lot more like ourselves. The second goal sort of made it a bit more difficult but I always felt we were a threat - unfortunately just not enough.
"We still scored, we had opportunities to create more chances. We were always still in the game but conceding an early goal and a second makes it hard."
Postecoglou continued: "Nothing really changes. We have a couple of weeks now where we don't have midweek games.
"This group has done an unbelievable job for two and a half months. I can't praise them enough, playing twice a week since November. They'll get the chance to reset now and finish the season strong.
"Europe is still very important to us, we're still in a great spot there, and we'll hope to get some players back over the next two weeks.
"We'll get players back which will help, we had 11 first-team players out today. Take that out of any team for one game and they would struggle - we've been doing that for two and a half months.
"The players are going out there and giving everything they can. It would be a lot better if they had some help."