The category five storm made landfall on the Camaguey Archipelago, off the north-eastern coast.
But the Bahamas have largely been spared after Irma changed track.
In Florida, 5.6 million people, or 25% of the US state's population, have been told to leave as the storm approaches. At least 20 people are known to have died so far across the Caribbean.
Irma, which strengthened in the last few hours, hit the Camaguey Archipelago late on Friday, threatening nearby coastal towns and villages.
This is the first time a category five hurricane has hit Cuba in decades.