Technically you should use a separate product that is formulated for the sensitive skin around the eye area. Face moisturizers are formulated for the skin on the face and therefore are not Ophthalmologist tested. This means they are not specifically formulated to work near the eyes and can cause irritation.
Also, an eye cream can help to target certain concerns you may have now or later in life. If you need to depuff your eyes or reduce under eye circles, a regular face cream will not help you achieve this.