The incident took place at Neelganj Road in Agarpara area.
According to police, 39-year-old Subrata Das alias Nepal used to stay with his parents and elder brother Gopal and his family. The brother-duo was involved in a property dispute.
Subrata attacked his sister-in-law Ratna (40) and nephew Subhankar (12) with a sword and slit their throats. When his wife Soma (34) threatened to commit suicide, he also slit her throat. The three of them died on the spot, police said, adding that his elder brother was not at home.
Subrata then tried to slit his 14-year-old daughter Shormistha's throat with the sword and attempted to strangulate his five-year-old son Somnath. However, they were rescued by the neighbours.
The two children were rushed to R G Kar Medical College hospital where their condition was stated to be critical.
Subrata has been arrested.