Melissa made landfall near the town of New Hope, 62km south of Montego Bay, packing maximum sustained winds of 295 km/h, the US National Hurricane Center said in its latest advisory.
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 destruction could be unlike anything people in Jamaica have seen before," US forecaster AccuWeather's hurricane expert Alex DaSilva said.
"The island has never taken a direct hit from a category four or a category five hurricane in recorded history."
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.
Shortly before landfall, Jamaican electric utility JPS said power outages had affected more than a third of its customers.
In its worst-hit parishes, three-quarters of customers lost power, JPS said.
Local Government Minister Desmond McKenzie told reporters that nearly 6000 people had moved into temporary shelters.
The government had issued mandatory orders for 28,000 people to relocate but some were reluctant to leave their homes.
"Don't bet against Melissa, because you will lose," warned McKenzie, as authorities implored residents to seek protection in shelters and fortify their residences.
"It's a catastrophic situation," the World Meteorological Organisation's tropical cyclone specialist Anne-Claire Fontan told a press briefing.
"For Jamaica, it will be the storm of the century for sure."
Storm surges of up to four metres were expected, she said, with rainfall set to exceed 70cm, causing "catastrophic flash flooding and landslides," she said.
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.
After crossing eastern Cuba, still as a powerful storm, Melissa is forecast to head through the Bahamas, where Prime Minister Philip Davis has ordered relocations for people in southern and eastern parts of the archipelago.
In Cuba, authorities said they had removed upwards of 500,000 people from areas vulnerable to winds and flooding.
"There are no half measures," Cuban President Miguel Diaz-Canel said in a message published in state newspaper Granma, in which he urged people in areas vulnerable to flooding, strong winds and storm surges to move and accept help.
"Melissa will arrive with force, and there's great concern about what it could destroy in its wake," he said.
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.
"Today will be very difficult for tens of thousands, if not millions of people in Jamaica," IFRC official Necephor Mghendi said via video link from Port of Spain in Trinidad and Tobago.
"Roofs will be tested, flood waters will rise, isolation will become a harsh reality for many."
To enable swift relief distribution, essential items - tarpaulins, hygiene kits, blankets and safe drinking water - had been pre-positioned in Red Cross branches on the island, he said, with more than 800 shelters set up.
Melissa made landfall in southwestern Jamaica, which was one of the areas worst-hit by last year's Hurricane Beryl.
with AP
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