Melissa made landfall near the town of New Hope, 62 km south of Montego Bay, packing maximum sustained winds of 295 km/h, the US National Hurricane Centre said in an advisory.
Some hours later, it had weakened to 241 km/h, the Centre said.
The slow-moving storm is forecast to remain a powerful hurricane as it crosses the mountainous island -whose highland communities are vulnerable to landslides and flooding -and heads towards Santiago de Cuba, Cuba's second-largest city.
The Miami-based hurricane centre warned that "total structural failure" was likely in Melissa's path.
"The island has never taken a direct hit from a Category 4 or a Category 5 hurricane in recorded history," said US forecaster AccuWeather's lead hurricane expert, Alex DaSilva.
"The destruction could be unlike anything people in Jamaica have seen before."
Melissa is the third most intense hurricane observed in the Caribbean after Wilma in 2005 and Gilbert in 1988, according to AccuWeather. Gilbert was the last major storm to directly hit the island.
Colin Bogle, a local adviser to aid group Mercy Corps in Portmore, near Jamaica's capital, said memories of Hurricane Gilbert run deep in the community.
"There is frustration that Jamaica continues to face the worst consequences of a climate crisis we did not cause," he said.
"Food aid will be needed, but recovery support like seeds, tools, and repairs for vehicles will be just as critical to help people restore their livelihoods."
Scientists warn that storms are intensifying faster as a result of climate change warming ocean waters.
Jamaica's electricity utility reported that more than a third of customers across the island had power knocked out.
Environment Minister Matthew Samuda told CNN his office was receiving videos of "severely damaged public infrastructure," including hospitals and places of safety.
Black River Hospital, 20 km southeast of where Melissa made landfall, reported significant damage to one of its blocks.
Local authorities reported before the storm made landfall that 6,000 people had moved into temporary shelters. The government had issued mandatory evacuation orders for 28,000 people, but some were reluctant to leave their homes.
In southeastern Jamaica, health authorities warned residents to watch out for crocodiles that could be displaced from swamps and rivers.
"It's a catastrophic situation," the World Meteorological Organisation's tropical cyclone specialist Anne-Claire Fontan told a press briefing, warning of storm surges up to four metres high.
"For Jamaica, it will be the storm of the century for sure."
Nearby Haiti and the Dominican Republic have faced days of torrential downpours leading to at least four deaths, authorities said. At least three people died during storm preparations in Jamaica, local media reported.
Melissa is forecast to cross eastern Cuba still as a powerful storm, then head through the Bahamas, where Prime Minister Philip Davis has ordered evacuations for people in southern and eastern parts of the archipelago.
In Cuba, authorities said they had evacuated about 500,000 people from areas vulnerable to winds and flooding.
"Melissa will arrive with force, and there's great concern about what it could destroy in its wake," Cuban President Miguel Diaz-Canel said in a message published in state newspaper Granma, in which he urged people in vulnerable areas to evacuate.
Melissa's slow movement over unusually tepid Caribbean water had contributed to its ballooning size and strength, NHC forecasters said, threatening Jamaica with days of never-before-seen catastrophic winds and rain.
The International Federation of the Red Cross said up to 1.5 million people in Jamaica were expected to be directly affected by the storm.
Jamaica's Prime Minister Andrew Holness has called for foreign support, saying the government had an emergency budget of $33 million and insurance and credit provisions for damage a little greater than that sustained from Beryl.
with AP
