In Tampa, which is affected by the warning, a small plane slammed into a house, killing one person, but authorities said it was unclear whether the severe weather already affecting the area played a role in the crash.
With the region still traumatised by 2005 hurricanes Katrina, Rita, Dennis and Wilma, the National Hurricane Centre predicted Alberto could turn into a hurricane and hit Florida's Gulf coast on Tuesday (local time).
"Preparations to protect life and property should be rushed to completion," said NHC forecaster Richard Pasch.
Local authorities ordered an evacuation of residents in threatened low-lying coastal areas of Levy County.
"Alberto has the potential to become a hurricane within the next 24 hours," said the US National Hurricane Centre, adding it could spawn tornadoes in Florida.
The hurricane warning was issued for the Gulf of Mexico coast from Florida's Longboat Key to the Ochlockonee river.
Cuban authorities earlier evacuated 28,000 people as the storm swept past the island packing winds of almost 70 miles (115 kilometres) an hour.
Eight people were injured, four homes were destroyed and 48 others damaged by a tornado at Nueva Paz, south of Havana, which was whipped up by Alberto, television reports said.
Air and sea transport to the Isle of Youth, south of Havana, was cut off by the storm, Cuban television reported.
Nine people have been killed in bad weather in Cuba in the past two weeks.
Alberto is the first storm since last year's record-smashing season of 28 storms, 15 of which became hurricanes.
Several of the hurricanes blasted across Florida, including the season's worst, Katrina, which left 1,300 people dead and tens of thousands homeless along the Louisiana and Mississippi Gulf Coast.
The city of New Orleans is still struggling to recover and engineers have warned its levees may not withstand another Katrina-style battering.
The hurricane season officially started on June 1 and lasts until November 30.
US weather experts are forecasting between eight to 10 hurricanes - six of them major - this year.