The accused, Ranjan Sethi (35), has been arrested and a sharp weapon used in the crime also seized, said inspector in-charge of Hinjili police station, Prasant Kumar Sahoo.
Previous enmity, coupled with the mental condition of the accused, might be the cause of the killing, according to preliminary inquiry by police.
The woman and her grand-daughter succumbed to injuries during treatment.
"We have registered a murder case and started an inquiry into the incident," Sahoo added.