Carvajal, a Spain international, had to come off five minutes from the end of the 2-0 victory in the Basque Country and Real said on their website (www.realmadrid.com) tests had confirmed the injury to his left thigh.
They did not say how long he would be sidelined for but local media reported he would be out for a month.
Real coach Rafa Benitez has been hit hard by niggling injuries to his players this term, with Gareth Bale, Karim Benzema, James Rodriguez, Sergio Ramos and Marcelo among players missing games.
Carvajal, who was sidelined for a month and a half earlier this season, looks likely to miss seven matches in all competitions before returning for the La Liga game at Valencia on Jan. 3.
Real play their King's Cup last 32, first leg at third-tier Cadiz on Wednesday.
(Reporting by Iain Rogers, editing by Ed Osmond)
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