The hurricane was centred 45 miles north west of Guantanamo, Cuba, and 205 miles south of the central Bahamas.
A hurricane warning was in effect for Cuba’s Granma, Santiago de Cuba, Guantanamo, Holguin and Las Tunas provinces as well as the south-eastern and central Bahamas.
Authorities said about 735,000 people remained in shelters in eastern Cuba.
“That was hell. All night long, it was terrible,” said Reinaldo Charon in Santiago de Cuba.
The 52-year-old was one of the few people venturing out on Wednesday, covered by a plastic sheet in the intermittent rain.
Parts of Granma province, especially the municipal capital Jiguani, were “under water”, said governor Yanetsy Terry Gutierrez.
More than 15in (40cm) of rain was reported in Jiguani’s settlement of Charco Redondo.
Officials reported collapsed houses, blocked mountain roads and roofs blown off.
There were no immediate reports of casualties.
Melissa was forecast to continue weakening as it crossed Cuba but remain strong as it moves across the south-eastern or central Bahamas later on Wednesday.
It is expected to make its way late on Thursday near or to the west of Bermuda.
