Thousands of tourists in the Florida Keys have also headed to safety on the US mainland.
Cuban authorities said they started evacuating tens of thousands of people from areas threatened by the storm.
In Florida, Governor Jeb Bush ordered a state of emergency, saying the south-eastern state "may be threatened by a major disaster".
Authorities ordered all visitors and non-residents in the Florida Keys to head to safety.
The Florida Keys - a chain of islands connected to the mainland by a series of bridges and a single road - are a major tourist destination.
The storm's projected track indicated Ernesto would cross Cuba, sweep over the Florida Keys, and eventually make landfall in western Florida.
After becoming a hurricane earlier in the day, Ernesto lost some of its punch and was downgraded to a tropical storm as it dumped torrential rain on Haiti and the neighbouring Dominican Republic.
At 0700 (AEST), it packed maximum sustained winds of 95 kph, according to the Miami-based National Hurricane Centre (NHC.)
Forecasters wary
But forecasters said it would likely regain power and reach hurricane status before it slammed into eastern Cuba.
Ernesto could strengthen further if it follows on its current track that would take it into the Gulf of Mexico one year after Hurricane Katrina devastated New Orleans and other coastal areas.
It could make landfall in western Florida on Thursday as a dangerous category three hurricane, packing winds of up to 209 kilometres and hour.
The NHC's five-day forecasts suggest Ernesto would spare New Orleans and other areas that suffered the brunt of Katrina.
But with memories of last year's devastation still fresh, US authorities were closely monitoring the storm's progress.
Hurricane Katrina, also a category three hurricane, struck the US Gulf Coast on August 29, 2005, killing 1,500 people in Louisiana and Mississippi and leaving hundreds of thousands homeless.
US authorities are keeping in close contact with the hurricane centre to ensure emergency officials are ready when the storm hits land.
Authorities urged residents of southern Florida to finalise "disaster plans" and store up on key supplies in readiness for the storm.
Forecasters said residents of Mexico's Yucatan peninsula should also keep a close eye on the storm's progress in case it deviated from its projected track.