Schools and businesses were shut and airport services suspended, as the cyclone made landfall with heavy rain and winds of up to 140km/h (85mph).
Teams from the army and National Disaster Relief Force, along with two naval ships, are on standby.
Fishermen in Andhra Pradesh and Tamil Nadu have been warned not to go to sea.
Police said disaster management teams had evacuated more than 15,000 people from low-lying areas.
The Tamil Nadu state government said in a statement that 260 trees in the city had fallen as a result of strong winds. Parts of the city have also been flooded.
Winds were so strong that glass panels were torn from the facade of Chennai's Hyatt-Regency hotel.
YK Reddy, director of the meteorology department in Hyderabad, told BBC Hindi the storm would weaken after making landfall.
The storm comes about a year after Chennai experienced devastating floods which killed 70 people.
In December 2011, more than 30 people died and many homes were damaged after a cyclone struck near the town of Pondicherry.
And in May 2010 at least 23 people died as a powerful cyclone caused widespread havoc across Andhra Pradesh.
At least seven people have been killed and thousands evacuated from coastal areas in two southern Indian states, as Cyclone Vardah lashed Chennai (Madras).