The 28-year-old midfielder returns to Turf Moor for an undisclosed fee - thought to worth up to £10m - having previously spent a spell on loan at the club between January 2010 and May 2011.
Cork now becomes Burnley's third signing of the summer having put pen to paper on a contract which will keep him at the club until 2021.
"It's great to be back. I've got some great memories from the year-and-a-half I was here and it's good to come back and see some old faces," he told the club's official website.
"It's good to come back and try and have another go here, especially in the Premier League. I've heard lot has changed, the whole set-up has changed and the playing staff is completely different to when I was here. It's a new challenge for me and one I'm excited for and I'm looking forward to meeting new people.
"[Swansea] had just signed someone in the same position, we had a lot of midfielders and it was going to be a difficult year. The bid came in and it was something I was keen on from the go. The club was keen and I was keen, we talked about it and decided it was best. It's been a busy couple of days, I had the medical yesterday and a bit extra today, flew out today and it's good to be here."
Cork leaves Swansea having made 83 appearances for the club since joining from Southampton in January 2015.