The storm had sustained winds of 40 mph (65 kph) and it was not expected to grow much further as it passes over the ocean east of Florida on a track that could carry it over parts of the Bahamas by evening, according to the U.S. National Hurricane Center in Miami.
It said the sprawling system could still cause heavy rains and spawn tornadoes in Florida, however.
By late Wednesday morning, the storm was centered about 120 miles (195 kilometers) east-southeast of Havana and 260 miles (420 kilometers) southwest of Nassau in the Bahamas. It was advancing toward the northeast at 9 mph (15 kph)
Cuba's chief meteorologist, Jose Rubiera, said the storm rolled across a swath of the west-central island overnight and its center was moving north of the island. Bands behind its core were continuing to bring heavy rains, however.
Rubiera said wind associated with the storm was not a threat, but that provinces from Matanzas east all the way to Guantanamo would continue to face downpours throughout the day.
"The important factor remains the rain," Rubiera said.
State-controlled television showed images of rain flooding roads and highways, especially around the eastern city of Santiago, but there were no reports of damage. Far to the west in Havana, it wasn't even raining and there was no flooding.
Communist Cuba has a well-trained civil defense force praised for its fast response to natural disasters, one that often uses mandatory evacuations to move people to safety in many parts of the island. Authorities often order thousands of evacuations ahead of even moderate storms - but there were no such orders reported for the depression.
Jamaica's Office of Disaster Preparedness and Emergency Management on Wednesday reported collapsed bridges, flooded roads and mudslides and it said that a boy was washed away before dawn when a house next to a paved gully collapsed in St. Andrew parish. Emergency workers were trying to recover his body from rust-colored waters.
Across the Caribbean country, several bridges collapsed overnight under the force of the flooded rivers and creeks. Schools and some businesses were closed as emergency officials braced for more rain through Friday.
In the capital of Kingston, underpasses flooded as the torrents overwhelmed storm water drains. Some motorists were stuck when their cars stalled in knee-deep waters. Most traffic lights were out and roads were littered with debris.
Police in Westmoreland parish's capital of Savanna-la-Mar said the community was hit by a waterspout overnight that ripped the roofs off a couple of buildings and sent four people to a local hospital with abrasions.
The depression was also felt Tuesday south of Cuba in Jamaica and the Cayman Islands, where meteorologists said more than four inches (10 centimeters) of rain fell in just 12 hours, causing flooding. Public schools closed and government workers from low-lying areas were allowed to leave early.
Chief Grand Cayman Meteorologist John Tibbetts said 5- to 7-foot (1.5- to 2-meter) waves were forecast through Wednesday night and warned boaters to remain ashore.
Associated Press Writer Tammie Chisholm in Grand Cayman, Cayman Islands, contributed to this report.