The 24-year-old striker arrived at the Liberty Stadium for the club record fee after agreeing his switch earlier this week.
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.
Bony is Swansea's third signing in 10 days, following Alejandro Pozuelo and Jonjo Shelvey in joining the Capital One Cup winners.
The 24-year-old was the leading scorer in the Eredivisie last season and netted 31 times in 30 games, earning him the Dutch Player of the Year title.
Since moving to Vitesse from Sparta Prague in 2011, Bony scored more goals than any other player in the Dutch league.
He told the club's 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, whose move is subject to receiving a work permit, hinted Swansea's place in the Europa League played a big part in tempting him to south Wales.
"The Europa League is my first focus," he said. "I enjoy playing in Europe. I am used to it.
"I am looking forward to joining the training camp out here in Holland as a Swansea player and building up good communications with the team and coaching staff as quickly as possible."
For Ivory Coast, Bony has won eight caps and scored eight goals, half of which have come in 2013.
Swansea's Dutch keeper Michel Vorm has no doubt Bony will be a big success in the Premier League.
He told Wales Online: "We're signing a player that will bring us goals.
"Swansea create lot of chances but there have been times we haven't scored as many as we should - but he will definitely improve us in that way.
"If you score 31 goals in Holland and don't play for Ajax or PSV or Feyenoord it is a huge achievement.
"It's a big coup and Swansea really fought for him but he really wanted to come here.
"I've spoken to him quite a lot since he met up with us and he's really happy to be signing, and I am 100 per cent sure he will fit in well."