Dejan Kulusevski's 90th-minute header earned a depleted Spurs a hard-fought point and ended their run of three straight defeats.
Spurs were missing 11 first-team players at the Etihad but led through Heung-min Son's breakaway goal inside six minutes before the South Korean turned a free-kick into his own net less than three minutes later.
City twice went ahead through close-range finishes from Phil Foden and Jack Grealish either side of half-time, but Spurs never stopped attacking, equalising through first Giovani Lo Celso and then Kulusevski.
"We're going through a tough time and I think how you deal with tough times is just as important [as results], I really do," Postecoglou said afterwards. "That's why I've been really strong on us not wavering from playing the football that we want to play.
"We have fallen short at times which considering the personnel out is understandable. If we can get through this period and still be in a decent position table-wise and get the players back and get some momentum and I do think that will happen.
"I'm confident it will happen unless it's going to be one of those extraordinary years where we're constantly with this attrition rate in our squad we can battle with the best.
"We're going through a tough period absolutely in terms of personnel, players playing in positions they're not familiar with and at the same time me not wanting them to compromise on the team we want to be so it's not easy.
"I always feel that if you can get through these tough periods in the manner which I think you need to and that is trying to stay true to what you started, trying to work on the team you want to become, you come out of it stronger, irrespective of the knocks you get along the way.
"I guess a day like today, if nothing else, gives the players a bit of belief in themselves as people more than players, that you know what, they do have that character and belief that even in the toughest of circumstances not to lay down."
Postecoglou admitted that City could have "blown us away" in the first half but felt his side were much improved after the interval, following the introduction of Pierre-Emile Hojbjerg for Bryan Gil.
"We hung in there and we made it uncomfortable for City I thought in the whole second half," he said.
"They never really got total control of the game and that's the key to getting something out of it. We scored three quality goals which you have to do so against a team like that. So pleased for the players that the effort they put in in the second half they got a reward for it.
"I thought Pierre was excellent in the second half. His experience out there, I thought we just as a group had a lot more conviction about our football, just passing it and not second guessing ourselves and being hesitant with our passes like we did in the first half."