Swansea's league double over United signals a momentous part of their history according to their captain.
The Swans stunned Old Trafford at the start of the season when they won by the same scoreline during Louis van Gaal's first game in charge at the club.
Williams told the South Wales Evening Post: "The manager spoke about it before the game, at half-time and at the end. It means a lot.
"Certain things you remember when you finish and I'm guessing this season and those two games will be right up there.
"When you consider the quality of manager they've got. For us, not such a big team as them, to win both games home and away means a lot to us.
"I think it shows the strides we've made this season. The first game of the season was massive for us. I look back on that with really fond memories and I always will."
The Swans appear to have made great strides under Garry Monk this season. Last term, the former captain was given the job as manager in order to stave off the threat of relegation following the sacking of Michael Laudrup, as the side struggled to juggle the demands of playing in the Europa League and the domestic competition for much of the season.
The side is on course to have their best season in the Premier League and wins against Manchester United home and away is one aspect which makes sides feel a bit uncomfortable at the prospect of coming up against the men from South Wales.