The two sides face off at St Mary's on Sunday, but Cork signed the contract after the midday deadline today so will not be able to feature for his new side.
Cork was in the final six months of his Southampton deal, and couldn't agree new terms with the Saints, despite boss Ronald Koeman saying an offer had been made to the midfielder.
Cork told Swansea's website: "It had come to the point where myself and the (Southampton) 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."