Cork finalised his switch to the Liberty Stadium after agreeing terms on a three-and-a-half-year deal on Friday, having fallen down the pecking order somewhat at Southampton.
The likes of Victor Wanyama, Morgan Schneiderlin, Steven Davis and James Ward-Prowse all provide tough competition for a midfield role at St Mary's Stadium, while academy graduate Harrison Reed is also emerging onto the scene.
Cork made just five Premier League starts under Ronald Koeman this season and is now looking ahead to life at Swansea.
"I'm glad to get it all done. It was tough because I'd been at Southampton for three and a half years and I had some great memories there," he told Swansea's official website.
"It had come to the point where myself and the club had gone as far as we could go. The opportunity of a new challenge at Swansea is something I’m looking forward to.
"Swansea are a good footballing team who have been very successful over the last few years.
"It's a team I've always looked at and admired, so for me to play in this team is a privilege.
"The style, the large core of British players and a manager who's hungry to do well made it an easy decision to come here.
"Every time I've played Swansea it has been a tough game. Earlier in the season it was a very hard match, while I played here [last season] under [Mauricio] Pochettino and we didn't touch the ball for the first 15 minutes of the game.
"It's always a very tough place to come, so it will be nice to be in the home side at the Liberty."
Southampton host Swansea in the Premier League on Sunday, but Cork will be unavailable due to the transfer not being completed before the requisite deadline.
