They said the bodies of the family, including children aged one, three and 15, were found in their house in Casteldaccia next to a small river which had burst its banks.
Separately a man was found dead in his car while two other people are missing.
Earlier this week in Sicily, floods closed many roads and mayors ordered schools, public parks and underpasses shut.
Italy, especially in the north and around Venice, has been hit by a series of strong storms which have claimed at least 20 lives and caused massive damage and disruption.
Trees covering the mountains in the Dolomites range were reduced to matchsticks, flattened by winds that tore through the Veneto region on Thursday.
"Thousands of hectares of forest were razed to the ground, as if by a giant electric saw." Italy's civil protection agency described the weather lashing the country this week as "one of the most complex meteorological situations of the past 50 to 60 years".
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