Three elderly people drowned when their retirement home near the city of Antibes was flooded.
Others died trapped in their cars in tunnels and underground car parks as the waters rose.
French President Francois Hollande announced a state of "natural disaster" in the affected region.
He thanked rescuers and expressed the "solidarity of the nation".
Mr Hollande offered condolences as he visited the retirement home in the town of Biot and urged residents in the region to remain cautious, saying: "It's not over."
Heavy rain hit the French Riviera on Saturday evening.
The city of Nice is estimated to have received 10% of its average yearly rainfall in two days alone.
The river Brague burst its banks, sending water coursing into nearby towns and cities. Social media pictures showed water gushing down the streets of Cannes.
Cannes resident Katya Higham-Stoianova told the BBC: "It was terrifying. We wanted to go out but decided not to as the rain was unbelievably heavy. The level of water was rising very quickly."
Cannes mayor David Lisnard said: "Some cars were carried off into the sea. We have rescued a lot of people, and we must now be vigilant against looting."
Eric Ciotti, president of the Alpes-Maritimes department, tweeted: "We have lived through an apocalyptic situation that we have never experience before."
The main motorway through the area has been closed, trains halted and hundreds of tourists sought shelter at Nice airport overnight. About 10,000 homes were still without power on Sunday evening, mainly in Cannes.
Some concertgoers attending a show by French rock veteran Johnny Hallyday at a venue in Nice had to sleep overnight there after becoming stranded.
Far-right leader Marine Le Pen's top aide sharply attacked her centrist opponent Emmanuel Macron, who is favorite to beat her for the French presidency, as campaigning for the final round on May 7 got under way within hours of first round results.
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