Ki has signed a three-year contract with the Welsh club, having passed his medical at the Liberty Stadium earlier today.
The exact fee has not been disclosed, although the club did confirm that Ki becomes the most expensive South Korean player in history.
The 23-year-old cited the Swans' style of play and manager Michael Laudrup as key reasons behind his decision to move from Scotland to Wales.
"When I look at Swansea I see a club that is growing and growing, and the style here is very attractive to me. 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," he said.
"It was very impressive, and I think I can grow as a player here and help the club move forward. The manager is another big attraction. I remember watching him when I was younger and I was amazed by how good he was. Now I want to learn from him in order to help me become a better player. I learned a lot at Celtic – it helped me become more of an all-round player.
"This is something I have dreamt about since I was a child – playing in the Premier League against the best players in the world. I can't wait to play my first game."
Ki, who won a bronze medal with South Korea at this summer's Olympics, will not feature in Swansea's match against West Ham United on Saturday but could make his debut against Barnsley in the League Cup on Tuesday.