Many had expected the Swans to still be relishing the celebrations of their victory over Bradford, and lacking focus when it came to meeting the Magpies.
But Luke Moore's late goal gave Michael Laudrup's side a 1-0 victory and moved them to 40 points in the Premier League table.
Defender Williams said: "It was always going to be a tough game in the circumstances after Wembley.
"They were very good as an attacking side. They hit the bar and forced Michel Vorm into some good saves.
"But I think that on the whole, if anyone deserved to win, it was us.
"Everyone had spoken about how we were going to react after the final on Sunday.
"Those people who thought we'd still be at Wembley, and still be in London, we wanted to prove them wrong.
"After we played Chelsea (in the two legs of the semi-final) we had positive results in both the next games and we did it again here."
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