A red alert asking residents to remain indoors was lifted on Friday.
But the cyclone caused widespread damage uprooting trees, damaging and flooding dozens of homes and severing power and water supplies.
The sole death from Thursday's cyclone Bejisa was an elderly person found outdoors, La Reunion's top government official Jean-Luc Marx said.
Two of those injured were in serious condition after falling from a roof and a ladder while trying to secure their homes, local officials said.
State-run power firm EDF said downed lines cut electricity to about 172,000 households.
Around 200,000 people - roughly a quarter of the island's population - had water cut to their homes.
The eye of Cyclone Bejisa passed within 15 kilometres (10 miles) of the western side of Reunion.
Waves more than eight metres (26 feet) high lashed the coast (as seen in the video below).
Several trees and traffic lights were uprooted on the island.
"The cyclone has not even struck and yet it has caused so much damage," said a resident from the western town of Saint-Gilles earlier on Thursday.
"I don't know what will happen when it's over us."
La Réunion suffered heavy damage in 2002 due to cyclone Dina, which claimed six lives and caused widespread flooding.
A man surveys flood levels in the Brittany town of Quimperlé on Thursday. Photo: D. Meyer/AFP
Back on the French mainland, three departments in Brittany were on alert for flooding on Thursday, after a storm hit the western region on New Year's Day.
A woman aged in her 50s was killed in Morbihan on Wednesday, when a tree uprooted by strong winds fell on the car in which she was a passenger, French TV TF1 reported.
The departments of Finistère, Morbihan and Ille-et-Vilaine remained on orange alert for flooding until 4pm on Thursday at the earliest, with water levels dangerous in several regional rivers, and a "high tidal coefficient" present in coastal areas.
The town of Quimperlé in Finistère experienced particular severe flooding on Wednesday and Thursday, as the river Laita overflowed on to the streets of the town.