Cyclone Sitrang crossed the Bangladesh coast near Barisal after skirting the West Bengal coast, the IMD said on Tuesday.
The weather in the southern West Bengal districts is likely to improve from forenoon, the India Meteorological Department (IMD) said.
The system, which moved towards Bangladesh from north Bay of Bengal at a speed of 56 kmph caused moderate to heavy rain and squally weather in West Bengal's coastal districts of South 24 Parganas, North 24 Parganas and Purba Medinipur, dampening festive spirits on Deepavali and Kali Puja.
Sitrang made landfall between Tinkona island and Sandwip near Barisal in Bangladesh between 9.30 pm and 11.30 pm on Monday with a sustained wind speed of 80 to 90 kmph gusting to 100 kmph, the Regional Met Centre in Kolkata said.
The Bangladeshi media reported that the cyclone caused heavy rain in southeastern parts of the neighbouring country.
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It is very likely to weaken into a depression and then into a low pressure by Tuesday evening, the IMD said.
The weather office warned of squally wind with speed reaching 40 to 50 kmph gusting to 60 kmph along and off West Bengal coast on Tuesday morning, which will decrease gradually to 30 to 40 kmph gusting to 50 kmph by forenoon.
Intermittent showers on Monday forced people to remain indoors as most Kolkata streets wore a deserted look on a festive evening which would normally have attracted tens of thousands to visit Kali Puja pandals and see the bright lights of the city.
The West Bengal government had advised people take necessary precautions in view of the weather forecast.