The gory crime occurred in a slum in state capital Bhubaneswar Friday when the man, whom the sisters shared as a husband, was away.
The matter came to light after the elder woman, a mother of three, rushed to the nearby police station and surrendered, offering the policemen also the knife she used to commit the murder.
Assistant Commissioner of Police (ACP) B C Mishra said the man, a daily wage labourer, married his wife's sister, then about 35 years old, about four years ago. The two women had been living with the man and the children as a domestic unit.
On Friday, a quarrel between the sisters threw the elder one into such a fit of rage that she slit her sister's throat, Mishra told IANS.
Police have arrested the woman on the basis of her confession, and sent the body of the deceased for autopsy.