Blot the oil with a paper towel immediately, if you catch the stain when it's fresh. Do not rub the paper towel, simply place it over the oil to absorb it, then turn the paper towel over to absorb any remaining oil on the surface.
Sprinkle corn starch directly on the oil stain. Let the starch sit for 15 to 30 minutes.
Brush the corn starch away using a soft brush, such as dry unused paintbrush. Alternatively, brush the starch away with a paper towel. If any oil stain remains on the silk pillow, repeat the corn starch application and removal until the stain is no longer visible.
Treat the spot with dry-cleaning fluid if it's still visible after several corn starch treatments. Moisten a cotton swab with the fluid, then rub the spot gently, avoiding contact with the silk surrounding the stain. Blot with a paper towel. Allow the spot to dry. Repeat until the spot is gone.