Archie Gray and Will Lankshear had Spurs 2-1 ahead after Gala's opener through Yunus Akgun. But the hosts stormed home through thanks to a superb double from Victor Osimhen.
Postecoglou later said: "Obviously disappointing result. First half wasn't great, you know, we just didn't handle things well at all, particularly with the ball, just really wasteful and gave it away way too many times, unnecessarily. That allows them to get a foothold in the areas that they're good at.
"They got some good players in the front third and we just allowed that to happen way too often and ultimately paid a price for it because I thought, you know we always finish strongly and I knew we would in the second half.
"Obviously going down to 10 men didn't help, but even with 10 men, I thought we probably played the best football, we did all game and got our second goal, had some opportunities to get a third there, but I thought we were looking the better side. So disappointing outcome, unfortunately, brought on by a disappointing first half.
"We expected a tough game here. Obviously, they've been in good form all year and undefeated at home. And, I thought the game was as expected. They have some dangerous players and you need to control the game with the ball better than we did tonight. I think that first half ultimately cost us the game and disappointing for us. Galatasaray are a good side. We know that. They're in a good position in the Europa League as are we. Hopefully that continues."