Winds of up to 180 kilometres per hour (110 miles per hour) and torrential rain caused extensive damage to the electricity network after cyclone Dumile passed within 90 kilometres of the island's west coast at around 0900 GMT.
EDF, the electricity supplier, said power to 68,000 homes had been cut.
An unspecified number of homes were also without water and the storm did extensive damage to avocado and other crops.
Jean-Luc Marx, the island's prefect, said 170 people had sought shelter in reception centres set up to house anyone concerned for their safety.
One fireman was slightly injured during an evacuation but otherwise no casualties were reported.
A red alert requiring all residents to stay inside remained in force in the early evening, although it was ignored by several thrill-seekers who received €135 ($177) fines for defying the order to witness the raging sea at first hand.
The cyclone was the first to hit Réunion since February 2007, when two people were swept to their deaths in an overflowing river.