"So far we have received information on three deaths. They all died due to falling trees," said Health Emergency Operation Control Room spokesperson Enayet Hussain.
Falling trees also caused the death of a woman and injuries to two other people in South 24 Parganas district in the Indian state of West Bengal, police sources confirmed to EFE news.
The police control room also reported that over the weekend they had established 69 shelters to accommodate almost 120,000 people from various districts of the state.
In Bangladesh, the number of evacuees exceeded 2.1 million, although some of them began returning to their homes, according to disaster management ministry secretary Shah Kamal.
"It's too early to say how much damage the storm caused. We have just started receiving information from different districts. As of now, my impression is that some 4,000 houses were damaged," he said.
In a special morning bulletin, the Bangladesh Meteorological Department said Bulbul "moved slowly north-westwards, weakened further into a deep depression and lies over Bagerhat, Barishal and Patuakhali (coastal) regions."
The Indian Meteorological Department maintained active rain alerts in the state of West Bengal, and in the northeastern states of Assam, Meghalaya, Tripura and Mizoram, all bordering Bangladesh.
It also warned of sustained winds of 85 kilometres per hour with gusts of 95 kph on the coast of Bangladesh and in the north of the Bay of Bengal, advising fishermen to leave their boats in port.
In Bangladesh, one of the worst-hit areas was Dakop in Khulna district, where the storm hit "early in the morning and an elderly woman died in a remote place surrounded by river as a tree fell on her house," local police chief Shafiqul Islam said.
Islam added that a primary assessment recorded 1,753 collapsed houses, with more than 5,000 families suffering other damages in the storm that lashed the area for about seven hours.
Ahead of the landfall on Saturday night, the Department had advised the maritime ports of Mongla and Payra to hoist 'Great Danger Signal No. 10'.
All fishing boats and trawlers in the North Bay have been asked to remain in shelter until further notice.
Bangladesh is prone to cyclones due to its geographical location. They occur mostly during April-May and October-November.
The last major cyclone to hit the county was Roanu, which killed 21 people in May 2016.
( With inputs from IANS )