Isla de la Juventud civil defense chief Ana Isla said there were "many people injured," but no reports of deaths. She said nearly all its roads were washed out and that some regions were heavily flooded.
"It's been very difficult here," she said on state television.
Authorities evacuated at least 300,000 people across Cuba, including western communities, cities near Havana and on the Isla de la Juventud, or Isle of Youth, an island of 87,000 people south of mainland Cuba.
Gustav was projected to plow into the Gulf of Mexico at full force Sunday, and reach the U.S. coast Monday afternoon. A hurricane watch was issued from Texas east to the Florida-Alabama border.
New Orleans Mayor Ray Nagin ordered the mandatory evacuation of the city, turning informal advice to flee from the approaching Gustav into an official order to get out.
More than a million Americans made wary by Hurricane Katrina took buses, trains, planes and cars as they streamed out of New Orleans and other coastal cities, where Katrina killed about 1,600 people in 2005.
City officials began putting an estimated 30,000 elderly, disabled or poor residents on buses and trains for evacuation ahead of Gustav's arrival.
Gustav already has killed 81 people by triggering floods and landslides in other Caribbean nations.
By late Saturday night, Gustav's eye was speeding over Cuba and was expected to soon reach the warm waters of the Gulf of Mexico.
The U.S. National Hurricane Center said the storm could then get even stronger, becoming a Category 5 hurricane with winds above 155 mph (249 kph).
Gustav was about 65 miles (105 kilometers) west-southwest of Havana and it was moving northwest near 15 mph (24 kph).
Cuba's top meteorologist, Jose Rubiera, said the hurricane's massive center made landfall in mainland Cuba near the community of Los Palacios in Pinar del Rio - a region that produces much of the tobacco used to make Cuba's famed cigars. There, the storm knocked down power lines, shattered windows and blew the roofs off some small homes.
Rubiera said the storm brought hurricane-force winds to much of the western part of Havana, where power was knocked out as winds blasted sheets of rain sideways though the streets and whipped angry waves against the famed seaside Malecon boulevard.
Felled tree branches and large chunks of muddy earth littered roads that were largely deserted overnight.
Cuba grounded all domestic flights and halted all buses and trains to and from Havana, where some shuttered stores had hand-scrawled "closed for evacuation" signs plastered to their doors.
Authorities boarded up banks, restaurants and hotels, and residents nailed bits of plywood to the windows and doors of their houses and apartments.
"It's very big and we've got to get ready for what's coming," said Jesus Hernandez, a 60-year-old retiree who was using an electric drill to reinforce the roof of his rickety front porch.
In tourist-friendly Old Havana, heavy winds and rain battered crumbling historic buildings. There were no immediate reports of major damage, but a scaffolding erected against a building adjacent to the Plaza de Armas was leaning at a dangerous angle.
Lidia Morral and her husband were visiting Cuba from Barcelona, Spain. She said Gustav forced officials to close the beaches the couple wanted to visit in Santiago, on the island's eastern tip earlier in the week. The storm also prevented them from catching a ferry from Havana to the Isla de la Juventud on Saturday.
"It's been following us all over Cuba, ruining our vacation," said Morral, who was in line at a travel agency, trying to make other plans. "They have closed everything, hotels, restaurants, bars, museums. There's not much to do but wait."
In the Gulf of Mexico, where about 35,000 people work staffing offshore rigs and production facilities, among other tasks, oil companies wrapped up evacuations in preparation for the storm.
As of midday Saturday, more than three-fourths of the Gulf's oil production and nearly 40 percent of its natural gas output had been shut down, according to the U.S. Minerals Management Service, which oversees offshore activity.
The U.S. Gulf Coast accounts for about 25 percent of domestic oil production and 15 percent of natural gas output, according to the MMS. The Gulf Coast also is home to nearly half the nation's refining capacity.
Analysts say prolonged supply disruptions could cause a sudden price uptick for gasoline and other petroleum products.
On Friday, Gustav rolled over the Cayman Islands with fierce winds that tore down trees and power lines while destroying docks and tossing boats ashore, but there was little major damage and no deaths were reported.
Haiti's Interior Ministry on Saturday raised the hurricane death toll there to 66 from 59 and Jamaica raised its count to seven from four. Gustav also killed eight people in the Dominican Republic early in the week.
Meanwhile, the hurricane center said Tropical Storm Hanna was projected to near the Turks and Caicos Islands late Sunday or on Monday, then curl through the Bahamas by early next week before possibly threatening Cuba.
As it spun over open waters, Hanna had sustained winds near 50 mph (85 kph) Saturday evening and the hurricane center warned that it could kick up dangerous rip currents along parts of the southeastern U.S. coast.
The U.S. State Department urged Americans to be aware of the risks caused by Hanna to people traveling to the Bahamas and the Turks and Caicos Islands. It urged U.S. citizens lacking safe shelter to consider leaving while flights are still available.
