Nathan Dyer and Jonathan De Guzman scored two goals each and Michu was also on target as they defeated their League Two opponents, who had goalkeeper Matt Duke sent off after 56 minutes.
Williams, skipper on the day in place of injured club captain Garry Monk, was delighted with the way his side held their nerve to claim the first major piece of silverware in the club's history.
"This is what we all dreamed about as little boys and the dream has come true, a final at Wembley," said Williams.
"We had a job to do and I thought we did it brilliantly today. We knew they were a bit of a dangerous team, they'd beaten Premier League teams before. We knew we had to do our job."
Williams admitted he did not feel the match was won until the fourth goal went in, when De Guzman scored a penalty in the wake of Duke's dismissal.
"The boys were really professional about it, they carried on playing and tried to score more goals," he added.
Williams went up the Wembley steps with Monk and the duo lifted the trophy together, a decision Williams revealed they had made in advance.
"We will lift the trophy together, we said all along we would lift it together, so that's a nice moment," he said.
Monk himself was delighted for the club, as they celebrated lifting their first major trophy in their centenary year.
The 33-year-old has been with the Swans since they were a League Two club and he felt it was due reward for all their hard work.
"It's about the journey we have been on as a club for so long. In our centenary year, to do this and bring a major trophy back to the club, it just puts the icing on the cake for the journey the club's been on.
"Swansea could have actually not had a football club, that's how close it was. To be on that journey and to come to where we are now, the board deserves so much credit. They have worked so hard behind the scenes.
"For all these fans here today, this is cup is for the club and the fans and the city itself."
Monk also praised manager Michael Laudrup for guiding them to some silverware at Wembley.
"He has brought us our first major cup in 100 years, so he deserves massive credit. We are hopefully on course for another top-10 finish.
"That's the pinnacle for a club like us, it's our second in the Premier League. If we can continue to progress and build, I am sure we can be there for many years to come."
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