The Premier League club said in a statement that Laudrup, who won four titles in a row with Barcelona and was also acclaimed as Denmark's best ever player, had signed a two-year deal.
Swansea's style last season paid tribute to a Barca side widely hailed as the best in the world and the Dane, who is expected to start work towards the end of next week, will keep up that tradition.
"Everyone knows the style of football Swansea play and it suits my way of thinking," said Laudrup, who has been working in Spain as a television pundit alongside former Swansea, Real Madrid and Wales manager John Toshack.
"It's going to be a new experience for me and I'm really looking forward to it," he told the club website.
It will be Laudrup's first foray into English football management, after resigning from la Liga side Real Mallorca in September, and he will be the third manager in four years at the Liberty Stadium.
Swansea finished 11th this year in their debut Premier League season, winning plenty of admirers with a possession style inspired by Barcelona's rapid passing 'tiki-taka' philosophy.
"We see Michael as someone who is fully committed and educated in all aspects of the way we want to take the club forward," said chairman Huw Jenkins.
"Our overall aim is to build on our first season in the Premier League, but we also fully understand the need to adapt our approach as we go along to remain competitive season after season," he added.
"Part of that process is to look at new ideas and increase our knowledge on all aspects of the game to keep advancing."
"I think he'll do well... I know Michael was very impressed with the way things are done [in the Premier League], the way in which he is in control of footballing matters, selling of players, buying of players," Molby told the BBC.
"I know that's some of the problems he's had at previous clubs and of course he has a very specific way he wants to play," added the former Liverpool midfielder.
"He was strongly influenced by Johan Cruyff when he
