"We spoke to the gaffer at the start of the season and we said we want to win trophies, as players and as a team," he told Spurs TV Online.
During the 29-year-old's Spurs career, he has been on the end of three defeats at the ground, FA Cup semifinal defeats to Portsmouth and Chelsea in 2010 and 2012 and a penalty shootout loss in the League Cup final against Manchester United in 2009.
Dawson missed the 2008 League Cup final win against Chelsea through injury so is hopeful of experiencing success in this season's FA Cup campaign.
"That's what you look back on at the end of your career. We have had two Wembley trips in the League Cup - one win and one loss," Dawson explained.
He added, "Wembley is a special place. To go up those steps and lift that cup would be special. And every player will say the same as me - the FA Cup is massive for us.
"We've had some good runs in it, we have been to Wembley twice, but had both unfortunate experiences against Portsmouth and Chelsea for different reasons. We want to go all the way and it would be a dream come true to do that."
The England international continued to state that his side must be at their best if they are to avoid a fourth-round exit against Leeds at Elland Road.
"It's a one-off game. If we are not at the races we will come unstuck," Dawson said.
"Leeds have some good players. It will be a sellout, the fans will be right behind them and the atmosphere will be rocking so we have to be ready. Neil Warnock always gets his teams up for them so we have to be ready and we certainly will be."