According to Swansea's official website, the fee is £12m plus add-ons, more than double the club's previous record of the £5.5m paid to Valencia to sign Pablo Hernandez.
Bony told the club website: "This move has been talked about for quite some time, so I am delighted it is all completed. Now it is important that I focus my mind on the challenge ahead because I hope to make the Swansea fans very happy.''
Bony was the leading scorer in the Eredivisie last season, with a remarkable 31 goals in 30 games. He could even be due for a reunion with his former employers soon, as Vitesse are in the same stage of the Europa League as Swansea.
"The Europa League is my first focus,'' added Bony, who finished the tournament's top scorer while at Sparta Prague with nine goals. "I enjoy playing in Europe; I am used to it.''