Karzai made a brief visit to Gurgaon on Wednesday to meet his wife and newborn daughter at the Fortis Memorial Research Institute.
Hamid Karzai has studied in India and graduated from Himachal university.
"The baby girl was born around 9.30 am on Tuesday. Both the mother and her baby daughter are doing quite well," Afghanistan's ambassador to India, Shaida Mohammad Abdali to TOI.
Abdali said the president paid a very short visit to the hospital during his brief transit stay in New Delhi before going to Colombo on Wednesday. "Both the president and the first lady enjoyed a few moments together with their newly born baby-daughter," the ambassador added.
Sources said Karzai's wife was admitted to the hospital earlier this week. She was accompanied by her family members and relatives. Karzai stayed with the family for around 20 minutes. The first lady is likely to be discharged by Friday evening, the sources added.
On choosing Gurgaon as the place for the delivery of their third child, Abdali said, "The president's two other beautiful children were born in Afghanistan. In this case, however, due to some predictable medical complication, the specialized doctor in Afghanistan advised that the baby be delivered in India given the availability of specialized medical facilities and personnel."
Abdali thanked New Delhi for its support. "We're thankful to the hospital's excellent medical staff for their superb services, as well as to the hospital's management team. And we're grateful to the Indian government for their continued support with everything, including facilitating the hospitalization process."
Both countries share co-operative relations with each other. India gave protection to Afghan's former defense minister Abdullah's family in Delhi, when he was fighting Taliban. They returned once the Taliban was defeated.