The national power company said strong winds caused the power system to fail.
At least 70,000 people were evacuated from their homes before the Category 3 storm hit on Wednesday evening, with warnings of storm surge, flash flooding and landslides.
This situation comes just weeks after millions of people were living without assistance electric current for four days, following power outages caused by problems with the country's energy infrastructure. The event also coincided with Hurricane Oscar which killed at least six people.
Rafael made landfall in the western province of Artemisa, near the capital Havana, at around 16:15 local time (21:15 GMT), the US National Hurricane Center (NHC) said.
A spokesman for the state energy company, Unión Eléctrica, said engineers would have to wait until the storm had passed before they could assess its impact on power plants and the power grid in general.
The NHC said the storm would continue to dump heavy rain in western Cuba on Thursday and storm surge could pose a danger to life.
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(source: AFP)