As a Category 4 storm, Gustav is already stronger than Hurricane Katrina was when it hit New Orleans in 2005, killing some 1,800 people and causing hundreds of billions of dollars in damage.
But US forecasters warned that the latest storm could yet strengthen further after it passes Cuba, potentially growing to a Category 5 - the highest possible classification - as it moves across the Gulf of Mexico.
After Cuba, Gustav's projected path takes it over the oil-producing Gulf, before making landfall in the US, possibly as early as Monday.
A hurricane watch has been put in place along America's North Gulf coast, from Texas along to the Alabama-Florida border.
In the US, the city of New Orleans has begun enforcing a mandatory evacuation order for coastal districts, or parishes, Emergency officials have warned people in vulnerable areas not to try and ride out the storm.
US Republicans say their convention in Minnesota next week could be suspended as the storm roars ashore.
John McCain, due to be nominated as the party's presidential election candidate, said the decision would depend on the storm's impact.
"It wouldn't be appropriate to have a festive occasion while a near-tragedy or a terrible challenge is presented in the form of a national disaster," he told Fox News.
Gustav hit mainland Cuba between in the west of Pinar del Rio province, reports said, having passed across the low-lying Isla de la Juventud, or Isle of Youth.
Predicted route of Hurricane Gustav (30 August 2008)
The island's civil defence chief, Ana Isla, said there were no reports of deaths but there were "many people injured".
She told the Associated Press that nearly all the island's roads were washed out and some regions were heavily flooded.
Cuban state media reported that many buildings were underwater and entire warehouses had been brought down.
The storm has already brought extensive flooding to the western, tobacco-growing province of Pinar del Rio and to Havana province.
Almost 250,000 people have been evacuated from low-lying coastal regions and half a million sacks of valuable dried cigar tobacco leaves were moved into safe storage.
The capital, Havana, is expected to escape the worst of the storm but its effects are being felt there - several parts of the city have lost power and large waves have poured over the Malecon, the capital's famous seafront promenade.
People rushed for cover as the hurricane lashed Cuba
The BBC's Michael Voss in Havana says Cuba, the only communist country in the Americas, has one of the best organised disaster-preparedness systems in the region.
Nevertheless, there have been reports of people fleeing from coastal areas carrying their belongings and valuable home appliances.
One woman said she was not afraid to leave home because "the police and authorities are always watching and taking care of things".
Storm surge fears
In the US, the Associated Press estimates that about one million people took to highways in the Gulf Coast region on Saturday to flee the storm.
Coastal parishes in New Orleans have been placed under mandatory evacuation, and long queues could be seen at assembly points in the city.
City mayor on Gustav threat
States of emergency have been declared in Louisiana and Texas.
Emergency officials have warned that a tidal storm surge up to nine metres (30ft) is possible along the coast.
David Paulison, head of the US Federal Emergency Management Agency (Fema), said the government had learnt from Katrina and was taking a "pro-active" approach to the storm, getting evacuation transportation and extra personnel on the ground early.
Gustav is the second major hurricane of the 2008 Atlantic hurricane season.
On Friday night, Gustav struck the low-lying Cayman Islands, flooding streets in the tax haven and luxury tourist resort but reportedly causing no injuries.
Earlier, the storm claimed the lives of at least 66 people in Haiti, eight in the Dominican Republic and seven in Jamaica, where heavy rains caused flooding and strong winds tore roofs off houses.
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