Rafael made landfall in the western province of Artemisa, near the capital Havana, around 4:15 p.m. local time (9:15 p.m. GMT), the US National Hurricane Center (NHC) said.
A spokesman for state-run 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 as a whole.
By 10:00 p.m., Rafael had moved into the Gulf of Mexico and weakened to a Category 2 hurricane with maximum sustained winds of 169 km/h.
The NHC said the storm would continue to bring heavy rains to western Cuba through Thursday and that storm surges could be life-threatening.
Heavy rain is also expected in the Cayman Islands, and a tropical storm warning is in effect for the Florida Keys.
