The Spain international has not featured since the game against Norwich on 15 December and Swans boss Michael Laudrup revealed on Christmas Eve Michu would have surgery.
Swansea confirmed the procedure has been a success and the 27-year-old is expected to return to action in February.
"Swans striker Michu has today undergone a successful ankle operation which will keep him out for six weeks," a Swansea statement read.
"Michu, who has scored six goals in 19 games this season, missed the festive fixtures with Everton, Chelsea and Aston Villa due to the injury.
"But after the operation went ahead as planned in Portugal, the 27-year-old is set to be sidelined for the next six weeks before making his return to the first team."
Michu, who has scored 28 goals for the Swans since joining from Rayo Vallecano in the summer of 2012, has been hindered by ankle and knee issues throughout the current campaign.