The 23-year-old Ki, capped 47 times by his country, has been with the Scottish champions since 2010.
The transfer fee was undisclosed, but believed to be a club record in the region of £5.5m.
He will not be part of the weekend's squad for the Premier League clash at home to West Ham United, but will be watch the game at the Liberty Stadium.
“When I look at Swansea I see a club that is growing and growing, and the style here is very attractive to me,” Ki told the Swansea website.
“Last year I watched them a lot in the Premier League and they did things very differently to a lot of the other teams. They wanted to keep the ball and pass it around, which I love to do. When they didn’t have the ball they worked so hard to get it back quickly.
“It was very impressive, and I think I can grow as a player here and help the club move forward.
"It will be unfortunate to lose a talented player but that's been our strategy at the club now for the last two or three years," Celtic manager Neil Lennon was quoted as saying on the BBC's website on Monday.
"We cultivate players then move them on. We feel it's good business."
Ki was one of South Korea's best performers as they won an Olympic bronze medal at the recent London Games.
Swansea City chairman, Huw Jenkins, said: “Having lost Joe Allen recently, it was vital that we invested the money wisely by bringing quality young footballers into the club with potential to grow further.
“Ki fulfils that criteria and everyone in the football club is excited about the prospect of him playing and developing with us."