Hurricane Oscar made landfall early on Sunday in the south-eastern Bahamas and was heading towards Cuba, an island recently hit by a massive power outage.
The National Hurricane Centre in Miami said the storm's centre arrived on Great Inagua island.
It is expected to produce a dangerous storm surge that could translate into significant coastal flooding there and in other areas of the south-eastern Bahamas.
Two to four inches of rain are expected, with isolated areas seeing up to six inches.
Forecasters said five to 10 inches of rain are expected across eastern Cuba through to Tuesday, with some isolated locations getting up to 15 inches.
Oscar formed on Saturday off the coast of the Bahamas and brushed past the Turks and Caicos islands to the south.
The National Hurricane Centre earlier described the storm as "tiny", but hurricane warnings were in place on Sunday for south-eastern Bahamas and portions of Cuba.
The storm's maximum sustained winds were clocked at 80 mph with higher gusts. Its centre was located about 150 miles east-northeast of Guantanamo, Cuba.
The storm was heading west at 12 mph and was expected to reach Guantanamo or Holguin, Cuba, on Sunday at hurricane strength.
The hurricane's approach comes as Cuba tries to recover from its worst blackout in at least two years, which left millions without power for two days last week. Some electrical service was restored Saturday.
Philippe Papin of the National Hurricane Centre said it was somewhat unexpected that Oscar became a hurricane on Saturday.
"Unfortunately the system kind of snuck up a little bit on us," Mr Papin said.
Hours earlier Tropical Storm Nadine formed off Mexico's southern Caribbean coast. It degenerated into a tropical depression as it moved over land.
