Seven infants were rescued from the blaze before firefighters arrived at the Bhandara District Hospital in the western state of Maharashtra.
India's Prime Minister Narendra Modi described the incident as a "heart-wrenching tragedy".
The cause of the fire is not yet known and an investigation is under way.
State health minister Rajesh Tope said initial evidence suggested it might have been caused by a short circuit.
The fire started at about 02:00 local time (20:30 GMT Friday), according to hospital officials.
Rescue efforts were hampered by continuous blasts inside the ward, local media report. One nurse on duty said she alerted the authorities after seeing smoke streaming out of the hospital's Newborn Care Unit.
"Hospital authorities have rescued seven babies but 10 have died in an unfortunate incident," district civil surgeon Pramod Khandate told reporters.
Some of the babies were only a few days old, and the oldest three months.
In a tweet, Mr Modi said his thoughts were with the bereaved families.
India's Home Affairs Minister, Amit Shah, said he was "pained beyond words" over the "irreparable loss".
Defence Minister Rajnath Singh said he was "deeply saddened" by the deaths, adding: "My heartfelt condolences to the bereaved families."
Bhandara is 62km (39 miles) from the north-eastern city of Nagpur - one of the largest cities of Maharashtra.