The Welsh outfit have failed to register a victory in their past seven Premier League outings, with their last success coming in a 3-0 triumph over Newcastle United on December 4.
Part of their downturn in form has been attributed to the fact Michael Laudrup's men have faced four sides in the Premier League's top six in recent weeks.
Williams believes he and his team-mates have applied themselves well in those encounters, but has called on the team to turn solid performances into victories to alleviate the threat of the drop.
Speaking to The Daily Star, he said: "We have to be concerned about relegation, so we need to get a win. We just need to learn a few lessons quickly and need to get a bit of experience in the critical moments.
"But we have to feel confident that what we've got in our dressing room is enough to take us well away from that. We knew we had a tough run against Chelsea, Manchester City, Manchester United and Tottenham.
"We've played well in all the games, but we're just not getting the result."
Swansea have also been affected by injuries this season, with the likes of Nathan Dyer, Pablo Hernandez and last season's top scorer Michu all enduring a spell on the sidelines.
But Laudrup is set to be boosted by the return of goalkeeper Michel Vorm, who is eyeing a return to action against Tottenham on Sunday after a month-long absence with a knee injury.
"You need to make sure you are 100 per cent and I am nearly there now," he told The South Wales Evening Post. "Hopefully I'll be fit to play against Tottenham at the weekend."
