Rafael made landfall in the western province of Artemisa, near capital Havana, at around 16:15 local time (21:15 GMT), the US National Hurricane Center (NHC) said.
A spokesperson for the state-owned energy company, Unión Eléctrica, said engineers would have to wait for the storm to pass over the island before they could assess its impact on power plants and the grid more generally.
By 22:00, Rafael had moved on into the Gulf of Mexico and weakened to a category two hurricane, with maximum wind speeds of 169km/h.
The NHC said the storm would continue to bring heavy rains across western Cuba through Thursday and that storm surges could pose a danger to life.
Heavy rains are also expected in the Cayman Islands, while a tropical storm warning is in force in the Florida Keys.
