Civil defense commander Sergio Simoes said five family members, including one infant and two adolescents, were among the dead.
In Baixada Fuminense, northern Rio, three rivers burst their banks, forcing around 200 people to flee from their homes.
Meteorologists forecast downpours to continue throughout the day, threatening New Year's Eve celebrations at the city's famed Copacabana Beach and elsewhere.
The authorities have called for residents in at-risk areas to move to safety, even if it means abandoning their homes.