The 26-year-old, who scored 15 goals in La Liga last season, came through his medical on Thursday and has put pen to paper on a three-year deal subject to international clearance.
He becomes manager Michael Laudrup's third summer signing following the arrivals of defender Chico Flores and midfielder Jonathan de Guzman. Laudrup believes capturing a player who had attracted the attentions of Manchester United, Liverpool, Sevilla and Valencia for such a low fee represents great business for Swansea.
He said: "It's a fantastic signing. We are talking about a player who scored 15 goals in the Spanish league last year for a small club who were saved by the bell in the last minute of the last game to avoid relegation.
"He was a big difference for them with his goals. We lost (Gylfi) Sigurdsson, who played in that position for half a season, so to sign a player like Michu is fantastic for us when you look at the price and quality of the player. It is a bargain.
"When you see the big newspapers in Spain and people commentating on how we can sign a player like that for such a small amount of money, then the Spanish economy must be really bad. It's a very difficult moment for clubs in Spain in general and this signing shows that."