Rafael, a Category 3 hurricane, ripped roofs from homes and bleachers from a baseball stadium as it barreled across the island, which was already reeling from a deadly storm last month.
The hurricane also caused a nationwide blackout, just two weeks after a power plant failure plunged the island into darkness for four days.
There were no reports of fatalities in the latest storm.
Nearly 250,000 people were evacuated from their homes before it hit, according to the authorities.
By Thursday afternoon, power had been restored "between the center and the east" of the island, President Miguel Diaz-Canel's office said.
The president visited affected parts of Havana, Artemisa and Mayabeque provinces, where plantain and yucca crops had been wiped out.
In the city of Havana, where 461 buildings partially or fully collapsed according to the authorities, residents used brooms, shovels and buckets to clear debris from the streets.
The highway from the capital west to the city of Artemisa was dotted with fallen electricity poles and trees.
In the town of Candelaria, around 24 miles (40 km) from where Rafael made landfall, 49-year-old housewife Lidia was in despair.
"Now, the hurricane is leaving and we have another blackout, meaning we won't have water," she said standing outside her house. "What are we going to cook? What water are we going to drink?"
Cuba has been suffering hours-long power cuts for months -- a symbol of the island's worst economic crisis since the fall of the Soviet Union, a key ally and financial backer, in the early 1990s.
The UN General Assembly last week renewed its long-standing call for the US to lift its six-decade trade embargo on the communist island.
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