Melissa hit the southern coast of eastern Cuba with maximum sustained winds of 195km/h, the US National Hurricane Center said.
"Life-threatening storm surge, flash flooding and landslides, and damaging hurricane winds are ongoing this morning," it said.
About 735,000 people were evacuated from their homes in eastern Cuba as the storm approached, authorities said.
Cuban President Miguel Diaz-Canel warned on Tuesday the storm would cause "significant damage" and urged people to heed evacuation orders.
Melissa had weakened to a still-dangerous category three hurricane after roaring ashore near Jamaica's southwestern town of New Hope on Tuesday packing sustained winds of up to 298kh/h, well above the 253kh/h threshold for category five, the highest level on the Saffir-Simpson hurricane scale.
In southwestern Jamaica, the parish of St Elizabeth was left "under water", an official said, with more than 500,000 residents without power.
"The reports that we have had so far would include damage to hospitals, significant damage to residential property, housing and commercial property as well, and damage to our road infrastructure," Jamaican Prime Minister Andrew Holness said on CNN after the storm had passed.
Holness said the government had not received news of any confirmed deaths from the storm, but "we are expecting that there would be some loss of life".
As daylight returned to Jamaica early on Wednesday, witness reports and videos on social media showed swathes of downed trees, washed-out roads and roofs tossed about fields and roadways.
Meteorologists at AccuWeather said Melissa ranked as the third-most intense hurricane observed in the Caribbean after Wilma in 2005 and Gilbert in 1988 - the last major storm to make landfall in Jamaica.
Scientists say hurricanes are intensifying faster with greater frequency as a result of warming ocean waters caused by greenhouse gas emissions.
Melissa's winds subsided as the storm drifted past the mountains of Jamaica, lashing highland communities vulnerable to landslides and flooding.
Local media reported at least three deaths in Jamaica during storm preparations, and a disaster co-ordinator suffered a stroke at the onset of the storm and was rushed to hospital.
Late on Tuesday, many areas remained cut off.
In the Bahamas, next after Cuba in Melissa's path to the northeast, the government ordered evacuations of residents in southern portions of that archipelago.
Farther to the east, the island shared by Haiti and the Dominican Republic had faced days of torrential downpours leading to at least four deaths, authorities there said.
The storm's centre, churning with violent wind gusts over 201km/h and heavy rain, slammed into Guama, a rural, mountainous area 40km west of Santiago de Cuba, the island's second-most populous city.
Authorities had shut down power to virtually all of eastern Cuba, evacuated vulnerable areas and had asked residents to shelter in place in the provincial capital Santiago, a city of 400,000 people.
Scarce videos posted by local media showed torrents of brown rainwater rushing down roads through dark towns at the base of Cuba's Sierra Maestra mountains not far from the city.
Authorities reported widespread flooding of lowland areas early on Wednesday from Santiago to Guantanamo, where more than a third of the population had been evacuated.
The timing could not be worse for the communist-run Caribbean island.
Cuba is already suffering from food, fuel, electricity and medicine shortages that have complicated life for many, prompting record-breaking migration off the island since 2021.
The hurricane was not expected to directly affect the capital Havana.
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