The Swans won their first major trophy on Sunday after beating League Two outfit Bradford City 5-0 at Wembley, 10 years after the Welsh side were nearly relegated from the football league.
However, Britton insists that the fans deserve the trophy for their support after they helped the club during their financial turmoil days in England's fourth tier.
"They passed round a bucket for my wages a few years ago and they deserve this day," Britton told talkSPORT.
"There's been some hard times at this football club but now, winning the cup is brilliant."
Swansea became the first Welsh club to win a major English trophy since 1927.