Named Alex, the storm brought fierce winds and torrential rain.
A number of villages north of Nice in France suffered serious damage from floods and landslides, with roads, bridges and homes destroyed.
In north-western Italy, flooding was described as "historic". A section of a bridge over the Sesia river collapsed.
French Prime Minister Jean Castex has deployed the army and released emergency funds to tackle the worst floods for decades in south-eastern France.
Up to 20 people are either missing or have not checked in with relatives.
"There are very many people of whom we have no news," Mr Castex said.
Bernard Gonzalez, prefect of the Alpes-Maritimes region, said: "Just because their loved ones haven't been able to get in touch doesn't mean that they have been taken by the storm."
He said the prospect of more rain was "a worry".
Meteorological agency Météo-France said 450mm (17.7in) of rain fell in some areas over 24 hours - the equivalent of nearly four months at this time of year, reports Reuters news agency.
The southern Alps region appeared the worst hit, with serious damage in the Roya, Tinée, Esteron and Vésubie valleys.
The villages of Saint-Martin-Vésubie and Rimplas were cut off, with roads inaccessible.