A midweek defeat after Sunday's painful loss to Arsenal would have increased the scrutiny on Postecoglou, who has repeatedly expressed his desire to win a trophy this season.
Tottenham are 13th in the Premier League ahead of Brentford's visit on Saturday, but the 58-year-old is no stranger to difficult times before silverware follows.
“You need to embrace the struggle. You don’t get success just by everything rolling out perfectly. There’s times when you’ve just got to roll your sleeves up and keep going," Postecoglou insisted.
“What’s the best process for me is to stay steely-eyed focused on what I think I need to do to get us to where we want to.
"I don’t think I’m in a unique space. I keep saying to people, show me a success story and I’ll show you a struggle. People forget the struggle and look at the end bit.
"You need to go through that. That’s the time that tests your resolve, tests your belief, tests everything you want to do"
Postecoglou flipped discourse about a potential crisis if they had lost to Coventry with a claim that he would be asked about the title if they had won at Leicester.
Johnson produced a 92nd-minute winner in the Carabao Cup after a difficult few days where social media abuse following the Arsenal defeat saw him deactivate his Instagram account.
Postecoglou said: “I haven’t spoken to Brennan about it. The lads know what I’m about and my beliefs.
"I keep telling them the same thing that the most important people in your life, they’re the ones you’ve got to worry about and their opinion.
“He won a game of football for us the other night with a really good finish at the critical moment. I reckon you put any of his critics in that situation and they would be looking for a change of pants pretty quickly."