The storm hit Patuakhali district on Thursday with winds of up to 100 kph (60 mph), and was heading for the ports of Chittagong and Cox’s Bazar.
One person has been reported dead, Bangladeshi officials say.
Thousands of people have been ordered to evacuate low-lying areas in Bangladesh and Burma, and take shelter in cyclone centres.
However, some displaced people in Burma have resisted calls for them to evacuate camps in Rakhine state.
The United Nations has warned that 8.2m people could be at risk from Mahasen in Bangladesh, Burma and north-east India.
The Bangladeshi authorities have raised the danger level to seven out of 10 for low-lying areas around Chittagong and Cox’s Bazar.
AdvertisementBBC Weather update on the progress of Tropical Cyclone Mahasen and the tracks it could take
However, Shamsuddun Ahmed, deputy director of Bangladesh’s Meteorological Department, told AFP news agency the cyclone was not expected to cause serious damage as it was “not severe”.
The cyclone “did not gain strength in the last part of its journey as it hit the coast”, he said.
In Bangladesh, there have been reports of waist-deep water submerging low-lying areas and houses being damaged. There are also fears of a storm surge.
All schools, colleges and some hotels have been declared cyclone shelters, the BBC’s Mir Sabbir reports from Dhaka. These centres are crowded and people are still rushing in, our correspondent adds.
Airports in Cox’s Bazar and Chittagong have been shut until the danger subsides.
‘Race against time’
In Burma, meanwhile, tens of thousands of Rohingya Muslims living in camps in low-lying areas of Rakhine state are feared to be at risk.
AdvertisementThe UN’s Kirsten Mildren says many in Burma are at risk from the cyclone
They were displaced by ethnic violence last year and many are reluctant to move from the camps.
Hla Maung said he lost his mother and two young daughters during the clashes between Muslims and Buddhists.
“I lost everything. I don’t want to go anywhere. I’ll stay here. If I die, I want to die here,” he said.
Rakhine state said it had moved some 36,000 displaced people from camps, Kirsten Mildren, from the UN Office for the Co-ordination of Humanitarian Affairs (Ocha), told the BBC.
But she said the evacuation was “not moving as fast as we’d like – it’s certainly a race against time. We’re finding it very difficult to convince [people] to move to higher ground or safer buildings.”
Burmese planning minister Tin Naing Thein claimed that in all more than 166,000 people had been relocated, but there was little evidence of a mass evacuation in reports from the affected area.
Correspondents say the Burmese evacuations are seen as a test of the government’s resolve to assist the Rohingya, amid allegations that state forces stood by or even participated in last year’s anti-Muslim violence.
On Tuesday, President Thein Sein was quoted as urging officials to “carry out relief work on humanitarian grounds for all, regardless of race and religion”, at a meeting to co-ordinate relief efforts for Cyclone Mahasen.
Cyclone Mahasen has already taken a toll. Though the storm did not make landfall in Sri Lanka, the associated heavy rain caused floods and mudslides which killed at least seven people, according to the country’s Disaster Management Centre.
At least 50 Rohingya Muslims were feared drowned on Tuesday when boats evacuating them from the path of the cyclone capsized off western Burma.