Postecoglou lifted the trophy after a 1-0 win against Manchester United on Wednesday night with Spurs also sitting in 17th place on the Premier League table.
He declared: "It's definitely been the toughest thing I have ever done, without a shadow of a doubt. And I knew that it would be going into it. This club has had some unbelievable world class coaches and players and they have not had a night light this, so I knew what I was taking on.
"It is fair to say that it has been really challenging because wherever else I have had success by this stage I have kind of had the faith and trust of everybody, and here it has been different. That's not a criticism, I understand that is because of the club and its recent history, I don't think they could always go in on one person. But for me that is always how it has worked well for me when people have backed my vision,
"Within that, when we got to the end of January, the end of the transfer window, I just made a decision there and then that this was the trophy that we were going to go for. It was probably at odds with what other people at the time thought we should do but I just really believed that we could win this.
"Since then every training and decision, everything we have done in terms of games, it was all about making sure when these games have come around we are in the best place possible. That has come at a cost in the league and I have to take responsibility for that. I juts felt like the end game of winning something was more important and it was the only way I was going to do it."
I'm not walking away
There had been claims of Postecoglou resigning even if Spurs won on the night, but he insists he's as a "hungry" as ever to continue the team's rebuild.
He also said: "Very (hungry). We've got a really young group of players. You can talk to them about success and what it means but until they feel it, it doesn't become real. I've got no doubt that all those boys tonight having this feeling they will want it again. To get it again they will need to make similar sacrifices.
"They have climbed the mountain now and know what it takes to get to where we are. I think that accelerates the opportunities to build a team and make it successful and make it competitive at the highest level for years to come.
"I think there is still a lot of work to be done, that's obvious, but not as much work as people may think. People may bang on about the 20 league defeats but they are missing the point of what we are trying to build here. I really feel that tonight can be a great platform for us to kick on."