The storm was expected to cross the coast early on Tuesday and slice diagonally across the island, entering near St Elizabeth parish in the south and exiting around St Ann parish in the north, forecasters said.
Hours before the storm, the government said it had done all it could to prepare as it warned of catastrophic damage.
Prime Minister Andrew Holness said: "There is no infrastructure in the region that can withstand a category five.
"The question now is the speed of recovery. That's the challenge."
Landslides, fallen trees and numerous power outages were reported ahead of the storm, with officials in Jamaica cautioning that the clean-up and damage assessment would be slow.
A life-threatening storm surge of up to four metres is expected across southern Jamaica, with officials concerned about the impact on some hospitals along the coastline.
Health Minister Christopher Tufton said some patients were relocated from the ground floor to the second floor, "and (we) hope that will suffice for any surge that will take place".
The storm was already blamed for seven deaths in the Caribbean, including three in Jamaica, three in Haiti and one in the Dominican Republic, where another person remains missing.
Melissa was centred about 241km southwest of Kingston and about 530km southwest of Guantanamo, Cuba.
The system had maximum sustained winds of 281km/h, according to the US National Hurricane Center in Miami.
"We will get through it together," said Evan Thompson, principal director at Jamaica's meteorological service.
Colin Bogle, a Mercy Corps adviser based near Kingston, said most families were sheltering in place despite the government ordering evacuations in flood-prone communities.
"Many have never experienced anything like this before, and the uncertainty is frightening," he said.
"There is profound fear of losing homes and livelihoods, of injury, and of displacement."
Matthew Samuda, Jamaica's water and environment minister, said he had more than 50 generators available to deploy after the storm, but warned people to set aside clean water and use it sparingly.
Melissa was also expected to make hit eastern Cuba late on Tuesday as a powerful hurricane.
A hurricane warning was in effect for Granma, Santiago de Cuba, Guantanamo and Holguin provinces, while a tropical storm warning was in effect for Las Tunas.
Up to 51cm of rain were forecast for parts of Cuba, along with a significant storm surge along the coast.
Cuban officials said on Monday that they were evacuating more than 600,000 people from the region, including Santiago, the island's second-largest city.
Melissa has also drenched the southern regions of Haiti and the Dominican Republic, with a tropical storm warning still in effect for Haiti.
The hurricane was forecast to turn northeast after Cuba and strike the southeast Bahamas by Wednesday evening.
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