Spence is set to start today against Newcastle after serving a one-match suspension.
Postecoglou said of his fullback: "He’s really knuckled down. He’s had to be patient. I think he’s been really good for us.
"Even in tough times you get these spurts of opportunities for others that maybe wouldn’t have happened. The reality is with everyone healthy maybe he wouldn’t have got the opportunity and in January we’d be looking at a different scenario for him.
"But right now, he’s performing really well. I was disappointed he got sent off against Forest because he could have helped us last week.
"He is very important for us. With Destiny going down, especially. Not because he's available but he’s played well, made an impact.
"Maybe he senses the opportunity to become not just part of Tottenham on a permanent basis in his own mind, where he feels settled, but now pushing to be a starting player and I think he has the capacity to do that."