Cyclone Mahasen has hit Bangladesh's low-lying coast, killing six people and forcing many thousands into emergency shelters.
Authorities downgraded warnings later in the day as the storm lost strength.
A storm surge did cause some flooding along the coast at high tide and thousands of rickety huts were destroyed by torrential rain and wind.
However, the devastation was not as bad as had been feared.
Neighbouring Burma, where there were fears for the safety of many thousands of internally displaced people living in camps, also appeared to have been largely spared.
Earlier, winds of up to 100km/h lashed the coast, whipping up big waves as the United Nations warned that 4.1 million people could be threatened.
A Bangladeshi army official at a control centre set up to help with relief work said six people had been killed.
Media said the death toll was ten, with some people killed by falling trees. About 50 people were injured, media said.
The storm was moving northeast, into northeastern India, as it lost strength, meteorological officials said.
Bangladesh, where storms have in the past killed many thousands of people, has more than 1,400 cyclone-proof buildings and many people moved into them as Mahasen approached.
But across its southeastern border in Burma, officially known as Myanmar, tens of thousands of people on the coast sheltered in camps and huts made of timber and palm fronds.
Burma's government had planned to move 38,000 internally displaced people by Tuesday from camps in Rakhine State in the west, most of them Rohingya Muslims who lost their homes in 2012 during violence between ethnic Rakhine Buddhists and Rohingyas.
Many had refused to relocate, afraid of the authorities' intentions, but they changed their minds after strong wind and rain last night.
At a camp near the state capital of Sittwe, a Reuters reporter saw Rohingya loading belongings into trucks provided by humanitarian groups, the UN refugee agency and the government.