Thousands of trees were felled by winds that destroyed homes and vehicles as hundreds of thousands fled to shelter homes.
Cyclone Gaja packed winds of up to 120 kilometres (75 miles) per hour as it barrelled into Tamil Nadu state after hitting the coast near Nagapattinam early Friday bringing heavy rains, India's weather office said.
The ruling party in the prosperous state said on Twitter that 11 people were killed in the powerful storm that has flooded several low-lying areas along the coastline on the Bay of Bengal.
The All India Anna Dravida Munnetra Kazhagam (AIADMK) party provided no further details about the victims.
The government has announced compensation of one million rupees ($14,000) to victims' families as hundreds of emergency workers were trying to restore roads and electricity.
The storm was predicted to move westwards and gradually weaken before heading into the Arabian Sea, the India Meteorological Department said.
Torrential rain caused waterlogging and threatened landslides. There were widespread power outages in the region.
Authorities closed schools and colleges and thousands of emergency workers were on standby along with two naval ships with divers, helicopters and inflatable boats.
Authorities had warned fishermen since Sunday not to risk going out to sea.
Gaja is the second major storm to hit India's east coast in recent weeks. Cyclone Titli battered Odisha state in October, killing at least two people.
Storms regularly hit southern India between April and December. Last year, Cyclone Ockhi left nearly 250 people dead in Tamil Nadu and Kerala.