By Agence France-Presse
Most were plucked from a sinking boat which had broken down, leaving them stranded at sea when Cyclone Mora hit on Tuesday, killing seven people and destroying thousands of homes.
The navy said its ships were conducting an "extensive search" of the area after the storm left a trail of devastation along the coast.
They are also carrying food, emergency relief and two medical teams to Kutubdia and Saint Martin's, two coastal islands that were hit hard by the cyclone.
"At least 15 ships have been deployed to search for survivors in the Bay of Bengal after the storm," a senior Navy official said.
"They are still in the sea and have rescued 23 fishermen from the water."
A navy helicopter and a maritime patrol aircraft are also conducting aerial searches for survivors, it said.
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.
However Mushtaq Ahmed, a fishing industry representative in Cox's Bazar, said eight boats carrying around 150 fishermen had failed to return.
"We heard some 60 fishermen were rescued by Bangladeshi and Indian navy ships. If they're our men, we think some 90 fishermen are still missing," he told AFP.
"They could be moored on a island or adrift at sea. In the past we have seen that most of the fishermen return to their fishing ports within a week or two. We hope this time they will also be back in good health."
Bangladesh suffers frequent cyclones and fishermen are often reported missing only to be found in their villages after the storm has subsided.
Ahmed said it was the first time the navy had conducted such an extensive search for missing fishermen after a cyclone.
The rescues brought the total number of people plucked from the sea since Tuesday to 56.
An Indian navy ship rescued 33 Bangladeshi survivors on Wednesday and handed them over to the local authorities, senior government official Zillur Rahman Chowdhury told AFP.
"All 33 are fishermen," Chowdhury said, adding they were tossed into the sea after their boats were caught in the cyclone.
Meanwhile, aid workers said emergency food and aid had reached tens of thousands of Rohingya refugees living in camps in Cox's Bazar, which bore the brunt of the storm.
The World Food Programme said it was handing out 100 tonnes of high-energy biscuits to nearly 20,000 refugee families after the cyclone destroyed or damaged thousands of homes.
"After a day of going hungry with my kids, this is amazing," said Rohingya refugee Abdul Khalek after he received 30 packs of biscuits for his family of eight.
Thousands of Rohingya have been sleeping in the open since the storm hit, despite continuing rain.
Cyclone Mora came after heavy rains in Sri Lanka caused the worst flooding the island has seen in well over a decade, killing more than 200 people.
South Asia is frequently hit by flooding in the summer with the arrival of the annual monsoon rains.