The defender enjoyed a solid second season for Paulo Sousa's side, making 22 appearances and scoring once as Swansea narrowly missed out on the play-offs.
Bessone joined in the summer of 2008 from Espanyol B on a free transfer and had a shaky first season in English football before improving vastly under the guidance of Sousa.
The Swans boss was keen to keep the left back, but despite a contract offer being on the table for several weeks, the Argentine decided to join Championship rivals Leeds.
Their boss Simon Grayson told the Leeds official site: "I watched him last year. He's very comfortable on the ball and is a very good acquisition.
"He has a good pedigree and he is another good player to come into the squad. We needed another left-back, Ben Parker was injured last year, and he gives us another option.
"He's a good age, he speaks good English, and he was part of a Swansea team who did well. He can handle a ball, he can defend, and he has done well when he has played in the Championship.
"We're delighted we've got him. He had an option to re-sign at Swansea which he turned down because of our interest in him.
"He's excited. He's aware of the club and the fanbase and having done well with Swansea last season he wants to go one better this season with us.
"He knows Luciano [Becchio] through Andrea Orlandi at Swansea.
"They have a similar background and hopefully he turns out to be a similar player."
This article first appeared on JACKARMY.net.