The baby had been in neonatal intensive care in Mumbai but died on Sunday.
The girl, who was 32 weeks pregnant, gave birth to the boy by Caesarean section on Friday, the hospital said.
Her father's colleague is under arrest for the rape. India normally allows terminations after 20 weeks only if the mother's life is in danger.
The case made headlines after her pregnancy was discovered on 9 August by chance when her parents took her to a doctor to seek treatment for suspected obesity.
A Delhi-based lawyer approached the Supreme Court on her behalf, seeking permission to abort the baby.
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In a landmark judgement, the court granted her permission on Wednesday to medically terminate her pregnancy.
The doctors had suggested a wait for two weeks to allow the foetus to grow further, but the judges ordered an immediate termination to avoid further trauma to the girl.
A panel of five doctors, headed by gynaecologist Dr Ashok Anand, carried out the Caesarean section at Mumbai's JJ Hospital.
Since her pregnancy was at such an advanced stage, the termination resulted in the birth of the baby boy, Dr Anand told the BBC on Monday.
"The mother is doing fine and we'll discharge her in a couple of days."
He said the baby boy had been was born slightly underweight at 1.8kg (4lb)
It's not clear what the baby died from, but a Times of India report quoting doctors said he had lung problems and severe breathing difficulties.
In another report, the newspaper said the "family had sprung a surprise showing its willingness to keep the baby".
Hospital staff told the BBC the 13-year-old had been informed about the infant's death and that so far she had been coping well.
The BBC's Geeta Pandey in Delhi says this is the third case in the past few months involving raped Indian children who became pregnant and sought the court's permission for a termination.
Last month, a 10-year-old rape victim gave birth to a baby girl in the northern city of Chandigarh. She too was 32 weeks pregnant, but she was denied permission to abort after a doctors' panel said that termination of her pregnancy would be "too risky".
In May, a similar case was reported from the northern state of Haryana where a 10-year-old, allegedly raped by her stepfather, was allowed to abort. She was about 20 weeks pregnant, doctors said.
None of the girls can be named for legal reasons.
These cases came to light so late because the children themselves were not aware of their condition and their parents also missed the obvious signs because they couldn't imagine that their daughters could be pregnant at such a young age, our correspondent says.
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