The 74-year-old Erramatti Mangayamma underwent a caesarean section after waiting nearly 60 years for children.
The couple married in 1962 and had been childless for 57 years.
Dr Sanakayyala Umashankar, director of Ahalya Hospital in Kothapet, said the surgery went “smoothly, both mother and infants are healthy and have no complications whatsoever.
“However, the mother has been taken to the intensive care unit to come out of the stress she had undergone for the last few hours.
“It’s another world record.”
Hours after their birth, thrilled dad Raja Rao hailed their arrival, saying: “Due to the grace of God and doctors, I am the proud father of two baby girls.
“We are the happiest couple on earth today.”
However, The Times reports he collapsed one day later and was taken to the Ahalya Nursing Home’s intensive care unit in Guntur, Andhra Pradesh.
The couple has remained in intensive care since their babies, who were conceived by in vitro fertilisation (IVF), were born on September 5.
The home has not revealed why the new mum remains in intensive care, and it has not given an update on Raja Rao’s condition.
The Hindustan Times says that because Mangayamma had no chance of producing eggs, as she had already gone through menopause 25 years ago, an egg was used from another donor, and fertilised with her husband’s sperm.
Dr Umashankar said: “Luckily she conceived in the first cycle itself, and she was found to be pregnant this January.”
He added that her condition had been “constantly monitored” during her pregnancy, including by cardiologists, gynaecologists and a nutritionist.
The doctor said: “Since it is very difficult for her to have normal delivery in view of her age, we had to perform a caesarean operation.”
Mangayamma’s brother, Devalla Raju, said the babies’ relatives are “all here for them, no matter what”.
She is three years older than another septuagenarian who gave birth in India after undergoing IVF treatment.
Dalinder Kaur gave birth to a boy, her first baby, in 2016 at the age of 72.
Spaniard Maria del Carmen Bousada de Lara delivered twins in December 2006 at the age of 66.