Officials had prepared for a cyclone, but the storm, called Mahasen, weakened considerably before making landfall.
The storm hit Patuakhali district on Thursday with heavy rain and wind of up to 100km/h (60mph).
Early reports suggest Muslim Rohingya living in camps on both sides of the Burma border were spared the worst.
The United Nations had warned that 8.2 million people were at risk from Mahasen in Bangladesh, Burma and north-east India.
Several Indian states issued storm alerts and warned people to take precautions against severe weather conditions.
The storm weakened over the Bay of Bengal, however, and forecasters say it is likely to dissipate within 24 hours.
The Bangladeshi government said it had evacuated 956,672 people from coastal areas to more than 3,200 cyclone shelters.
Officials broadcast warning messages before Mahasen hit.
Airports in Chittagong and the resort town of Cox’s Bazar were closed, and Chittagong’s port also remains closed, says the BBC’s Sanjoy Majumder in Chittagong.
The Bangladeshi authorities earlier raised the danger level to seven out of 10 for low-lying areas around Chittagong and Cox’s Bazar.
The cyclone covered more than 175km in nine hours before hitting the coast, Bangladesh’s Meteorological Department said.
A storm surge destroyed thousands of huts and caused flooding in coastal areas.
All schools, colleges and some hotels have been declared cyclone shelters, and most were packed overnight.
‘Race against time’
In Burma, there were fears for tens of thousands of Rohingya Muslims living in camps in low-lying areas of Rakhine state.
They were displaced by ethnic violence last year and many were reluctant to move from the camps.
One refugee, Hla Maung, said he had lost his mother and two young daughters during the clashes between Muslims and Buddhists last year, and would not move from the camp.
“I lost everything. I don’t want to go anywhere. I’ll stay here. If I die, I want to die here,” he said.
Burmese Planning Minister Tin Naing Thein said that in all, more than 166,000 people had been relocated.
In the event, the storm changed course and appears to have caused only minor damage in Burma.
Cyclone Mahasen earlier hit Sri Lanka, causing floods and mudslides that killed at least seven people, according to the country’s Disaster Management Centre.
At least 50 Rohingya Muslims drowned on Tuesday when boats evacuating them from the path of the cyclone capsized off western Burma.