The INS Sumitra found the survivors, including women and children, floating in the Bay of Bengal roughly 160 kilometres off the coast of Chittagong.
"Our navy have found 33 survivors and one dead body from the sea. They were floating in the waters when they were found," Dhaka's Indian High Commission spokesperson Ranjan Mondol said.
The Indian embassy's Facebook page showed pictures of survivors floating in the Bay of Bengal as well as of rescued people covered in blankets.
Eastern Naval Command spokesman Commander CG Raju said the rescued Bangladeshis were given medical treatment.
Cyclone Mora smashed into Bangladesh's coastline around Cox's Bazar district early Tuesday, packing winds up to 135 kilometres per hour.
Officials said the cyclone killed at least seven people, including the person found by the Indian Navy, and destroyed or damaged more than 20,000 homes. Nearly 6 lakh residents have been evacuated from coastal areas.
As the cyclone advanced on Bangladesh, local authorities increased the maritime threat to its highest level and ordered all fishing vessels to remain at port.
"We think they could be stranded or have anchored in far-flung islands or villages," association secretary Abdul Khalek told AFP.
In the past many fishermen reported missing during cyclones return to their villages after the storm has subsided.
The Bangladeshi navy has said it was sending ships to assist the tens of thousands affected by the cyclone, and to comb the seas for any stranded survivors.
"Our ships are also actively working on search and rescue operations in the region," said Taposhi Rabeya, a spokesperson at the armed forces division.
Get Breaking news, live coverage, and Latest News from India and around the world on NDTV.com. Catch all the Live TV action on NDTV 24x7 and NDTV India. Like us on Facebook or follow us on Twitter and Instagram for latest news and live news updates.