Three health officials have been suspended over the incident at the Guntur Government Hospital in Andhra Pradesh state where the 10-day-old boy was being treated for a urinary tract problem.
His mother, Chavali Lakshmi, said she alerted staff after noticing some of her son's fingers appeared to have been nibbled off during his stay in the neonatal intensive care unit.
"Our son was about to undergo a surgery but we noticed wounds on his right hand on Wednesday. Rats had nibbled them (fingers) off," Lakshmi told TV reporters.
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Andhra Pradesh Chief Minister N. Chandrababu Naidu said he has suspended three officials and ordered an inquiry into the incident.
"Shocked and deeply disturbed after learning that a baby boy died after bitten by rats in Guntur hospital," Naidu said on Twitter.
"Warned officials that such an incident should not recur."
The hospital's top official, medical superintendent T Venugopala Rao, could not be contacted for comment despite repeated attempts.
But he was quoted in the Times of India newspaper as saying the boy's death occurred despite putting up rat mesh in the neonatal intensive care unit after receiving complaints about rodents.
"It is an unfortunate incident," he said.
Most of India's 1.2 billion-strong population rely on under-staffed and run-down government hospitals for free treatment.
Last year, a hospital in the central state of Madhya Pradesh hired pest controllers after incidents of rodents damaging hospital equipment and biting patients.
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