Seven others from the woman's family are suspected of involvement in the crime, police said, adding that raids are being conducted to nab them.
The bodies of Dharmender, 22, and Nidhi, 20, victims of "honour killing", were found Wednesday.
Dharmender's headless body was thrown outside his house in Garnavathi village near Kalanaur, about 250 km from Chandigarh, and 80 km from the national capital, Wednesday evening. The girl's body was cremated by her family in secret.
Police took possession of Dharmender's body, but were only able to recover charred remains of the woman.
The woman's father, Billu, has been arrested. Both the victims' families have fled the village since the crime, police said.
Police suspect that the girl's family had asked the couple to return from Delhi. The family was unhappy that she was having an affair with the young man, and had eloped with him Tuesday.
The victims were friends studying in institutions in Rohtak, 10 km from their village.
The Khap panchayats (community councils) of this region disallow marriages within the same village and "gotra" or caste lineage. Although the khap decisions do not have the sanction of law, they are influential in many village communities.
In the past, young couples have been killed and attacked by khaps which do not accept marriages within the same gotra or village.
In some cases, the man and woman, even after they get married, are forced to accept each other as brother and sister.