The U.S. National Hurricane Center (NHC) said Rafael made landfall in the western province of Artemis near the capital Havana at approximately 16:15 local time (21:15 local time).
A spokeswoman for state energy company Unión Eléctrica said engineers would have to wait until the storm passed the island to assess its impact on power plants and the power grid more generally.
By 22:00, Rafael had moved into the Gulf of Mexico and weakened to a Category 2 hurricane with maximum winds of 169 km/h.
The NHC said the storm would continue to bring heavy rain across western Cuba through Thursday and that storm surge could put life at risk.
Heavy rain is also expected to fall on the Cayman Islands, and a tropical storm warning has been issued for the Florida Keys.
