Routledge told the club website: "I'm looking forward to the challenge.
"It's taken a couple of days to get everything agreed, but I'm excited about getting started."
Routledge says he is eager to work with Swans' manager Brendan Rodgers having admired the south Wales' club's attractive style of play.
"I've had a good chat with the manager and he really impressed me," he said.
"I had heard a lot of good things about him and the chat only reinforced my belief that moving to Swansea was the right thing to do.
"Having watched Swansea last season, I think everyone would agree they were the best footballing side in the Championship.
"It's going to be a difficult first season in the Premier League; it's going to be tough. But everyone is confident and looking forward to it. After all, playing in the Premier League is what you dream of.
"I'm excited about it."