The Swans beat Chelsea in the semi-finals of the competition to book their chance of glory next Sunday, although the 30-year-old midfielder remembers when the Welsh side were going through more difficult times.
Britton reveals that the players used to have to train on beaches and in gardens and was too embarrassed to tell manager Michael Laudrup he won the LDV Vans Trophy in 2006 - but says a win at Wembley would make it all worth it.
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“But it’s funny because [Michael Laudrup] asked us in a team meeting the other day, 'Has anyone played in a final?' Me, Garry Monk and Alan Tate all wanted to put our hands up, but we were too embarrassed.
"You can't say to Michael Laudrup, 'Yeah, we played in the LDV Vans final against Carlisle and won 2-1' - he's won La Liga titles and things, so we kept our hands down.
"It was huge for Swansea when we won the LDV Vans. We all thought, 'This is it, we're in a final, it's at the Millennium Stadium and it’s on the telly'. But, no disrespect, I doubt too many people remember who has won the LDV Vans.
"You would have never thought that, seven years down the line, we would be in a Capital One Cup final at Wembley, with the chance to win a major honour and qualify for Europe."
Britton admits it was hard to adjust to Swansea's meagre set-up after joining from West Ham in 2003, with the club lacking even the most basic facilities of a football club.
"It was a bit of a culture shock at first," he continued. "I had been at Arsenal and West Ham, with famous players, big grounds and great training facilities.
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"The players used to take their own kit home to wash, I ruined loads of kits and lost them a few times. We had to take our own drinks to training as well."
After such experiences Britton feels a win over Bradford would be a sign of what the club has achieved during his time there, and a bonus for everybody involved in the recent success at the Liberty Stadium.
"Some of the lads at Swansea now wouldn't believe me, but it's nice to know exactly how far the club has come," he added. "This is Swansea's centenary year. To get to a major final for the first time is brilliant and to win it would be incredible. It would be fitting for everybody who has been part of the transformation."
The Swans must first overcome Liverpool at Anfield in the Premier League on Sunday before turning their attentions to silverware, but will be buoyed as they beat the Reds away en route to Wembley this season.