The attack came as the Himalayan region disputed by India and Pakistan holds its first elections in 27 years to choose town councils.
Muslim separatists fighting against Indian rule have called for a boycott of the polls, which are held over several days.
The grenade attack on Sunday night took place in a remote village in Doda district, southeast of Srinagar, the summer capital of Indian-administered Kashmir.
"A group of militants stormed the house of Abd al-Aziz around midnight and lobbed grenades. Four family members died on the spot in the explosions," a police officer said.
Abd al-Aziz and another daughter were wounded, the officer said, adding that it was not yet known why the family was targeted.
"A group of militants stormed the house of Abd al-Aziz around midnight and lobbed grenades"
Unnamed police officer
Separatists have in the past attacked police informers or villagers who have refused to shelter them.
Although violence continues in Kashmir, Indian authorities say the levels are lower than in 2003, coinciding with moves by New Delhi and Islamabad to make peace.
India accuses Pakistan of stoking the revolt in Kashmir, the cause of two of three wars between the arch rivals. Pakistan denies the allegations.
Leaders of the two countries are due to hold talks on the sidelines of a regional summit in Bangladesh next week.