While the woman is under treatment in hospital, the child was declared dead on arrival, a Defence press release said.
The 24-year-old sailor, Vishnu Unni of INS Sharda, saw the woman and child falling into the water while he was passing by the Vikrant Venduruthy Bridge.
Braving strong currents and in total disregard of his personal safety, he successfully put both the woman and the child into a boat of the Cochin Shipyard limited (CSL) which had reached the spot by then.
The sailor is a native of Palakkad in Kerala.
The woman and child were taken to a private hospital, where the child was declared dead on arrival, it said.