The 24-year-old striker arrives at the Liberty Stadium for a club record fee of £12m after agreeing his switch earlier this week.
"This move has been talked about for quite some time, so I am delighted it is all completed," he told the club's official website.
"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 named Dutch Player of the Year last term after he netted 31 goals in 30 games for Vitesse as the club finished fourth in the Eredivisie.
And Bony admits he is looking forward to Europa League football with the Swans, where he could face his former club at some stage of the competition.
"The Europa League is my first focus,'' added Bony, who finished as the tournament's top scorer while at Sparta Prague with nine goals. "I enjoy playing in Europe; I am used to it.
"I watched Swansea a lot last season on TV. They have some good players and I like the way they play. I appreciate the style, which is similar to the Dutch way, and I think being at Swansea will help me develop as a player.
"I believe it's the right step for me; the perfect move. I had a lot of offers from all across the world, including England, France, Ukraine, Russia and the UAE.
"But I chose Swansea because it is a good club with a corresponding style, a great manager and, of course, playing in the Premier League. I considered everything and Swansea came out on top.
"I had my best season last year at Vitesse with 36 goals in all competitions. It is going to be tougher in the Premier League to score goals, but I will try my best. Everything is possible.
"This move has been talked about for quite so I am delighted it is all done. First I need settle into the training camp and build up good communications with the team and coaching staff."