There were no reports of any human casualties.
"Yes there have been tremors of the earthquake, which has its epicentre in Sumatra," said G.C. Debnath, director, Regional Meteorological Centre, Kolkata.
While the first tremor was felt at 2.08 p.m., the second one occurred about an hour later.
Hundreds of people ran out of their homes and offices and gathered on the streets. The tremors were felt in various parts of the city, especially in Lake Town, Salt Lake and Park Street area.
Two multistoried buildings in central Kolkata - one at Bowbazar and the other on Lindsay Street- developed cracks, a police officer said.
As a precautionary measure, metro services were suspended for 48 minutes between 2.42 p.m. and 3.30 p.m.
"All Metro trains were vacated at different stations and passengers were requested to leave the stations premises immediately," a Kolkata Metro Railway statement said.
The aftershocks were felt from Darjeeling in north Bengal, to East Midnapore, North 24 Parganas and South 24 Parganas in the southern extreme of the state.
"Like the rest of the country, an alert has been issued in the coastal areas of our state. We have alerted the district magistrates, the Block Development Officers and police officers. I also appeal to the people to be cautious," Chief Minister Mamata Banerjee told mediapersons after arriving in the city this evening from Delhi.
While the India Meteorological Department (IMD) put its intensity at 8.5 on the Richter scale, the US Geological Survey classified it an 8.9 temblor, whose epicentre was at a depth of 33 km.
The earthquake had its epicentre off the west coast of northern Sumatra. There have been no reports of loss of life or property in India or elsewhere.
A tsunami alert was issued for 28 countries, including India, Sri Lanka, Australia, Myanmar and Bangladesh.