It was expected to steadily lose strength as it moved across the island before hitting the central Bahamas by Sunday night or Monday.
Early reports of damage in Cuba were limited, but state television said the late-season storm toppled a major communications tower on the southern coast and interrupted electricity and phone service.
In the central-eastern province of Camaguey, more than 220,000 people were evacuated from low-lying areas. Another 170,000 people were moved in the eastern province of Las Tunas.
Former President Fidel Castro warned in an essay published by Cuban state media Saturday that Paloma would damage roads and crops planted after hurricanes Gustav and Ike hit in late August and early September, causing about $9.4 billion in damage and destroying nearly a third of the island's crops.
In the city of Camaguey, 79-year-old Rosa Perez waited out the storm at a government shelter with her 83-year-old husband and about 900 others from the town of Santa Cruz del Sur.
Perez was a toddler when she watched her mother, older sister and about 40 other relatives swept away in a storm surge during a 1932 hurricane that killed about 3,000 people.
"We're just waiting to see what happens to our home and our beach," she said.
Fellow Santa Cruz del Sur resident Aida Perez, who is not related, watched the news with her daughters, ages 19 and 10.
"This is a really hard blow," the 44-year-old said, adding she was certain they would lose their home and everything in it. "What's important is that we are alive."
Outside on the nearly deserted, flooded streets, four men struggled in pouring rain to carry a refrigerator to a more secure building.
Late Saturday, Paloma was located about 35 miles (60 kilometers) south-southeast of Camaguey. Once packing winds of 145 mph, the storm had begun to weaken over land and was moving northeast at about 7 mph (11 kph). It was expected to slow Sunday as it crossed Cuba and hit the open Atlantic by late morning.
Still, hurricane force winds extended up to 30 miles (45 km) from the storm's center and rainfall was due to reach 5 to 10 inches in central and eastern Cuba, with isolated totals of 20 inches possible.
"Although it may weaken a bit, we have to pay full attention to this storm," top Cuban meteorologist Jose Rubiera told state television and radio.
Before Paloma made landfall, Cuba's National Information Agency said crops, poultry and pork operations were being protected in the eastern provinces of Camaguey and Santiago. State television showed workers warehousing bags of rice, trimming tree branches and clearing storm drains. Bus and train service was suspended across central and eastern Cuba.
Evacuations were not reported in other provinces, but Cuba regularly relocates masses of people to higher ground ahead of tropical storms and hurricanes, preventing major losses of life. The hurricane center said Paloma could bring parts of the island battering waves and a life-threatening storm surge of up to 23 feet (7 meters). Rubiera, the Cuban meteorologist, warned residents along the southern coast to be especially vigilant.
Elsewhere, Paloma knocked out power across much of Grand Cayman island, downing trees, flooding low-lying areas and ripping off roofs. But residents appeared unscathed as businesses reopened and electricity and water service were restored on Saturday.
Donovan Ebanks, chairman of the Hazard Management Committee, said no injuries were reported.
"There has been minimal, if any, damage on Grand Cayman," Ebanks said.
Paloma's fierce winds ripped the roofs off some buildings on Cayman Brac, to the east.
A tropical storm warning remained in effect late Saturday for the central Bahamas, including Cat Island, the Exumas, Long Island, Rum Cay, San Salvador, Acklins Island, Crooked Island and the Ragged Islands.
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Associated Press writers Anita Snow in Havana and Trent Jacobs in George Town, Cayman Islands, contributed to this report.
