Andre Ayew's late goal snatched a 2-1 win for the Swans who had not won in their previous six games in all competitions
Ayew's brother Jordan had given Villa the lead before Gylfi Sigurdsson levelled with a fine free kick at Villa Park on Saturday
Swansea's victory marked the end for Villa boss Tim Sherwood as he was sacked on Sunday following a run of six straight defeats
The win lifted Garry Monk's side up to 11th in the Barclays Premier League and Wales defender Williams believes it will give them a base to climb the table.
"We know we can play better than we did, but it was scrappy and we had to dig out a win - sometimes you have to do that," he told the club's official site.
"We are not happy with some areas of our play but, as captain, I am happy from the response to the other night (a 1-0 defeat to Stoke).
"It's a really nice feeling
I said to the lads that someone else has to have the horrible feeling of losing as we'd not won for a while.
"Now it's about trying to build on this result
We need to get some consistency.
"So the message will be to use this result to push on and not be up and down in terms of performances and results."
Source : PA
Source: PA