"The little girl survived due to the heroic gesture of her grandfather," the Livorno Telegrafo newspaper reported. The family was asleep when the water flooded into their small flat and had no time to react.
Divers worked to recover the bodies as tearful neighbours watched.
Tourists had to change their plans as Livorno railway station closed and a nearby campsite was evacuated.
"The city is on its knees," said Filippo Nogarin, the local mayor. The Government had underestimated the danger, issuing a code orange alert for the region rather than red, he added.
"We didn't expect this because the alert was orange. Then we woke up to this," he said, adding that the death toll "may still rise" but could have been avoided entirely.
Italy's civil protection service said the code orange alert for Florence was still in place as the storms, which began in northern Italy, swept southwards down the country.
Underpasses were closed in parts of Rome, and flooding caused seven metro stations to close. The city council advised residents to stay at home and to avoid parks until they had been checked for possible fallen trees.