More than one month's rainfall fell in just a couple of hours as powerlines faltered and darkened large parts of the city during the overnight cloudburst.
A river broke its banks and traffic jams brought much of Athens to a standstill in the morning rush hour.
Police said a 27-year-old woman died of a probable heart attack after being trapped in her car in the northern suburb of Halandri.
"This is the worst storm since 1961. We're talking about 52 years without ever having seen such a heavy downpour in this area," meteorologist Yannis Kallianos told state television.