Brennan Johnson had Spurs ahead before Tom Cairney found an equaliser for the visitors. Cairney was also sent off late on.
Postecoglou later said: "It was a tough game, tight game, not a lot of openings for either side. We started well, should have been 1-0 up, then they had their spell and Fraser (Forster) made good saves. We just could not push on. We tried to get momentum late in game when they were down to 10, but just couldn't get it done.
"I know people use (the win at) Man City as a reference point, but I was able to put on Johnson and (Timo) Werner, had (Guglielmo) Vicario in goal. I know people are looking for easy targets, but what the players are having to deal with is self evident.
"We're going to have to protect our players. We don't have many options, so will manage them the best we can.
"I've had questions about Man City game, in seven days we have copped some significant blows, we can't play like we did against City in every game. With this club, everything we do well is used as a millstone to bring us down at every opportunity. We can't react to any of that. I just feel, the players we have at the moment are working hard to get us where we want to be, and we are still in decent shape, decent position in the league."