The Swans reached the final mostly thanks to a 2-0 win at Stamford Bridge in the first leg of the semi-final.
"The LDV Vans trophy in 2006 was a massive, massive game for Swansea at the time," Britton said. "But it's funny because [Laudrup] asked us the other day 'Has anyone played in a final?'
"Me, Garry Monk and Alan Tate were too embarrassed. You can't say to Michael Laudrup 'Yeah, we won the LDV Vans final against Carlisle United 2-1', he's won La Liga titles and things.
"Some of the lads now wouldn't believe me [about conditions at the club ten years ago]. 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."
Swansea take on Bradford at Wembley next Sunday.