Britton was a key player in their 5-0 thumping of Bradford City at Wembley.
He told The Sun: “It is still hard to believe what Swansea have achieved. It is now only just sinking in. But not everything was straightforward.
“Last Thursday at our Press day, myself, Michu, Jonathan de Guzman and Dwight Tiendalli were all asked to give urine samples at 1pm.
“But I was too hydrated and my sample was clear. It took me until 4.30pm to give a sample that was acceptable.
“After the final and that amazing result at Wembley, I was asked to give another urine sample. I couldn’t believe I was asked again. I took my own magnum of champagne in there. I also took two bottles of beer.
“I sat next to Bradford’s Gary Jones. I had just won, coming in on cloud nine and he had just lost. So we had a beer each. But I still took 90 minutes to give a proper sample as I was so dehydrated.
“I ended up a little bit drunk. By the time I got to the changing rooms everyone had gone. I was gutted. I got in there hoping a few people would be around but there was just the kit lady and a physio. I missed holding the cup and all the good pictures.”