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 (www.swanseacity.net).
It will be Laudrup's first foray into English 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.
He will also be the first Dane to manage in the Premier League.
Laudrup, who celebrated his 48th birthday on Friday, replaces Brendan Rodgers who moved to Liverpool last month with three backroom staff. Swansea said his assistants would be confirmed at a later date.
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.
The Dane will bring the experience he acquired as a Barca and Real Madrid player who won five successive Spanish titles between 1991 and 1995 with a reputation as an elegant passer of the ball.
"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."
The attacking midfielder also won league championships in Italy and Netherlands with Juventus and Ajax.
He won 104 caps for Denmark and has managed in Denmark, Spain and Russia.
Former Swansea player-manager and compatriot Jan Molby said earlier in the week, when the appointment was looking a done deal, that Laudrup would be an ideal appointment.
"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 was at Barcelona in the late 1980s, early 90s.
"That's how he wants to play his football that's how Swansea played last year under Brendan Rodgers, so if it happens I think it's a marriage made in heaven."