Human rights activists hold a protest over honour killings.
The girl's father and brother opposed her affair with a lower-caste boy.
According to police, the 14-year-old confessed on Sunday after her arrest to mixing poison with a vegetable curry that she served to her father and younger brother who both died after consuming the dish.
The incident occurred on Saturday in Malpur village, 75km from Patna, the state capital of Bihar.
The girl's mother, Sunita Devi, told police she survived because she had been ill and refused to eat the food.
"The girl took this extreme step as her family members, including father, brother and mother, were … against her love affair with a boy and her plans to marry him," senior police officer Upendra Kumar Sinha said.
The boy was a member of India's Dalit community.
'Untouchables'
Dalits, or "untouchables" as they used to be known, occupy the lowest rung in India's rigid caste hierarchy.
Most live in poverty and do menial, supposedly "unclean" jobs like collecting garbage.
The girl's family, which belong to a higher caste, had previously beaten the boy when the couple were caught trying to elope.
Such incidents resulting from disputed inter-caste relationships more commonly see male family members murdering their daughter or sister for bringing "dishonour" on the family.
There are no official figures on honour killings, though an independent study in 2010 suggested as many as 900 were committed every year in the northern Indian states of Haryana, Punjab and Uttar Pradesh.
Bihar is one of India's most impoverished and caste-ridden states.