Police say she is in a critical condition after being admitted to hospital on Sunday. They have arrested the 28-year-old cousin.
Delhi Commission for Women chief Swati Maliwal, who visited the girl, described her injuries as “horrific”.
The debate over sexual violence in India has grown after the fatal gang rape of a female student in 2012.
The rape of the baby girl happened on Sunday but came to light on Monday after local media reported it.
Ms Maliwal tweeted that the baby had undergone a three-hour operation and that her cries could be heard in the hospital.
This distressing case of assault on an infant has shocked India and made national headlines. The extent of her injuries has horrified many and prompted them to wonder whether we have reached a new low.
But a look at the statistics, compiled by the government, shows that such crimes are not uncommon.
And worryingly, their numbers are rising rapidly.
According to the latest National Crime Records Bureau data, 2016 saw 19,765 cases of child rape being registered in India – a rise of 82% from 2015 when 10,854 cases were recorded.
A couple of years ago, an 11-month-old was kidnapped by a neighbour while she slept next to her mother and brutally raped for two hours.
And in November 2015, a three-month-old was kidnapped and assaulted in the southern city of Hyderabad.
How are people reacting to the news?
Ms Maliwal described what she heard when she visited the hospital where the baby was being treated.
“Her heart-rending cries could be heard in the intensive-care unit of the hospital. She has horrific injuries in her internal organs,” she tweeted after visiting the hospital on Monday night.
She posted another tweet expressing her anguish.
“What to do? How can Delhi sleep today when 8 month baby has been brutally raped in capital? Have we become so insensitive or we have simply accepted this as our fate?”
She also tweeted a direct appeal to Prime Minister Narendra Modi that “stricter laws and more police resources” were needed to protect girls in the country.