Just before the storm made landfall on Wednesday, an oil tanker with 37 crew ran aground off Chennai. One of its lifeboats capsized in the choppy waters and one crewmember drowned, the Press Trust of India (PTI) news agency reported.
On Thursday, two helicopters searched in the Bay of Bengal for the four missing crewmembers, said coastguard officer Gurdip Singh, adding that 32 crewmembers were rescued from the tanker.
Andhra Pradesh state said two people died there when their homes collapsed due to heavy rain on Wednesday night in Nellore and Chittoor districts, and PTI reported another death in Tamil Nadu state, a 46-year old man who slipped into the rough sea from a pier and drowned.
Sri Lanka reported two deaths earlier from the cyclone.
The cyclone had maximum winds of 45 mph after landfall but had weakened to a tropical depression. A storm surge of up to five feet was forecast to flood low-lying coastal areas, the India Meteorological Department said.
Power supply was disrupted to parts of Chennai, the capital of Tamil Nadu state, PTI said.
State authorities turned 282 schools into relief centres in Chennai and the city's port halted cargo operations. Twenty-three ships were moved to safer areas. About 150,000 people were moved to shelters in Nellore, district official B Sridhar said.
In Sri Lanka, 4,627 people were displaced by flooding and 56 fled because of a landslide threat in the island's central region.
One woman died on Tuesday after a tree branch fell on her while another person was killed in flooding, the nation's Disaster Management Centre said. Floods also damaged about 1,000 houses, it said.