The "severe cyclone" brought heavy rains and winds with speeds up to 120km/h as it made landfall some 50km north of the state capital, Chennai.
"Two people died in the initial hours of the storm, including a girl who was killed when her house collapsed," an official at the state disaster management control room said.
Hundreds of trees and electricity poles have been uprooted. An estimated 17,000 people are being evacuated to relief camps in Tamil Nadu and its neighbouring state of Andhra Pradesh.
Schools and businesses were shut and Chennai airport services suspended as heavy rain and winds lashed the city.
Met officials later said Vardah - which means red rose - has gained in speed and intensity on its journey across the Bay of Bengal.
On its way inland, Cyclone Vardah hit several islands in the Andamans last week, uprooting trees and electricity poles besides severely impacting tourism in the archipelago.
