The skin under your eyes is much thinner than anywhere else on your body, and for that reason it may have trouble retaining moisture. For a small amount of dry skin, a common solution is to use some aloe on the area. Some people go so far as to use Vaseline. If you are looking for a more natural alternative to petroleum jelly, there is Beeseline.
Using your clean pinky finger or a cotton swab, apply a small amount of “jelly” to the affected area.
If the area in question is very close to the eye, to avoid getting any product in your eye,
you can apply it to the nearby area and let the skin absorb moisture as needed.

If one of these options has cured your problem, excellent. However the dry skin could come back so it is worth adopting a natural routine to keep the sensitive skin healthy.
1) Stay hydrated – drinking the correct amount of water a day – 2 Litres or 1/2 a gallon will help keep your skin soft
2) Eat a healthy diet that consists of adequate fruits and vegetables.
See the new Canada Food Guide.
3) Wash your face with warm or cool water – do not use hot water on the face
a) make sure that you fully wash away your cleanser, residue can create dry patches
b) dry your face by patting it with a clean towel, rubbing can cause irritation
4) Use a natural eye cream, ideally one that contains aloe or cucumber
a) apply around the ocular bone, as the skin in the area will extract the moisture that it needs. As
the area is very thin-skinned and therefore gentle applying moisture directly to the area can cause it to
become oily.
5) Try an exfoliant once a week – a very gentle or mild one to avoid irritation
6) Avoid excess cosmetics and product

If the dryness has not cleared up within a few days, you may want to seek the advice of a medical professional.