The Spanish international opened the scoring in the 28th minute when he collected a pass from Juan Mata, before slotting the ball into the bottom corner of the net.
The 24-year-old went on to claim his second goal of the afternoon with 15 minutes remaining, scoring from inside the area after collecting a pass from Jose Bosingwa.
The newly-promoted Swans did grab a consolation four minutes from time when centre-back Ashley Williams headed in Mark Gower's free-kick.
There was still time for ten-man Chelsea to add a fourth, with Didier Drogba making a goalscoring return from injury.
Chelsea travel to Bolton Wanderers next Sunday afternoon, while the Swans host Stoke City on the same day.