The victims included a one-year-old baby and children aged three and 15.
Three other members of the family managed to escape, one by climbing a tree, the Agi news agency reported.
Six Italian regions remain on high alert for storms.
After visiting the stricken Mediterranean island by helicopter, Italian Premier Giuseppe Conte said the other victims included a German couple whose vehicle was swept away by flood waters near Agrigento, a tourist town known for Greek temples.
In the same region, a 44-year-old man was also found dead in his vehicle near Vicari after he had tried to reach a trapped colleague in a service station. A 20-year-old passenger in the vehicle with him is still missing.
Heavy rains and gale-force winds have been battering Italy for several days, uprooting millions of trees and cutting off villages and roads.
The tragedy brings the number of people killed in Sicily this weekend to at least 12 after three other people died in their cars when hit by torrents of water.
The severe weather has caused massive damage and disruption. Trees in mountainside forests in the northeast of the country were flattened like matchsticks.
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"It's like after an natural disaster", Zaia said.
Earlier this week, floods in Sicily had closed many roads and mayors ordered schools, public parks and underpasses shut. "Thousands of hectares of forest were razed to the ground, as if by a giant electric saw". Troops were deployed Sunday to check the condition of the main roads.
On Sunday, after flying over the region with Italy's Interior Minister Matteo Salvini, Zaia said the storms had destroyed 100,000 hectares of pine forest in all.
Mr Salvini, who is also Deputy Prime Minister, said a rough estimate of how much it would cost to safeguard Italy against such events was 40 billion euros ($63.45 billion).
The bloc has objected to Italy's proposed budget, which it says will worsen the country's already huge deficit.
The canal city of Venice, on Italy's northeast coast, has also experienced some of its worst flooding ever, as well as having to withstand winds of up to 180 kilometers an hour (110 miles an hour).
Italy's civil protection agency described the extreme weather the country has been enduring as "one of the most complex meteorological situations of the past 50 to 60 years".