Chico reunited with Laudrup at Lekhwiya
Laudrup was appointed as the new coach of the Qatari champions last month, having been sacked by Swansea in February.
And he will now be joined at the club by Chico, who released an open letter to Swansea's fans on Saturday as news of his move was confirmed.
"I write this letter with grief, gratitude and joy, for I will be facing a new personal and sporting challenge," wrote the 27-year-old.
"I have experienced so much these two seasons that, by far, exceeded my expectations. That is why today I am sad to leave a club that has been like a family since the beginning and allowed me to enjoy and experience football in a very special and intense way.
"I was so honored [sic] to be part of a roster with not only great players but people that have become great friends. And having the chance to contribute to the winning of the Capital One Cup and to bear the name of Swansea throughout Europe in the Europa League, is so historic and will remain as a memory in us all. I leave Wales with a lot of friends."
Chico's time at Swansea, following a move from Genoa in July 2012, was not without controversy, with the Spaniard involved in a number of on-field flashpoints.
He added: "A very important thing I wanted to say is that if, at any time, I disrespected any opponent, coach, referee or rival fans I apologise. Football is a sport that we live with the pulsations very high and sometimes we may make mistakes, but the important thing is to learn from them.
"I go to another country and another league. Here in Europe I enjoyed three of the best leagues in the world ... and I have experienced so much. Hence, the new challenge I face will be given the same enthusiasm as what I came with to Swansea two years ago.
"I hope this is not a forever goodbye. I will support you all from afar; I will always be a Swan."