Matthew has been downgraded to a Category 1 hurricane by the US National Hurricane Center.
Hurricane Matthew was about 35km south-east of Charleston, South Carolina as it moved northeast at 19km/h, the center said in an advisory on Saturday morning.
Matthew had maximum sustaining winds of 140km/h, it said.
The storm system has killed nearly 900 people, including at least 877 in Haiti.
Matthew reduced much of the Cuban township of Baracoa to rubble, whipping up giant waves that demolished cement buildings and winds that tore off roofs - but it didn't take lives.
Largely thanks to a rigorous evacuation scheme, Cuba managed to avoid the fate of neighbouring Haiti, where nearly 900 deaths have been reported so far in the wake of the strongest storm to hit the Caribbean in nearly a decade.
"If we had stayed here, this would have killed us," said Aristides Hernandez, 76, who abandoned his apartment overlooking the sea for a friend's house inland.
"The sea came in through the front and exited out the back and took everything with it," he said.
Upon returning home, he found his cement balcony ripped off by the waves and washed through. The wave also took away his doors, windows and the wooden stairwell.
Hernandez said volunteer civil defence members in the city near the eastern tip of Cuba went door to door, advising residents to evacuate.
Cuban state TV ran storm advisories on a loop and officials blared warnings from vehicles with loudspeakers.
"The civil defence didn't rest ... Thanks to that, no one died," Hernandez said.
The death toll in Haiti in the wake of Hurricane Matthew has soared to almost 900.
Tens of thousands in the impoverished Caribbean nation have also been left homeless by the monster storm, which is now ploughing northward over waters just off the US southeast where it has caused flooding and widespread power outages.
The number of deaths in Haiti surged to at least 877 on Friday as information trickled in from remote areas previously cut off by the storm, according to a tally of death tolls given by officials.
Matthew triggered mass evacuations along the US coast from Florida through Georgia and into South Carolina and North Carolina.
There were at least four storm-related deaths in Florida but no immediate reports of significant damage although the storm swamped streets, toppled trees and knocked out power to more than a million people.
Two people were killed by falling trees, according to Florida officials, and an elderly couple died of carbon monoxide poisoning from a generator while sheltering in a garage.