Tropical storm Elsa brought drenching rain and strong winds to Cuba Monday but left the island without major damage, as the US National Hurricane Center reported the storm was now making its way toward the Florida Keys.

Elsa has left a trail of destruction through the Caribbean, claiming at least three lives, and is expected to gain strength as it emerges off the coast of western Cuba Monday evening.

Over 100,000 people were evacuated from coastal or low-lying areas as the storm cut through the island, with Cuba's meteorological institute Insmet reporting winds of up to 100 kilometers an hour.

By 9 pm (0100 GMT), it was headed out toward the Gulf of Mexico, authorities said.

But Florida appeared set to avoid the worst of the storm as forecasters shifted its likely path westward, suggesting the southwest coast of the state will experience a glancing blow rather than the direct hit earlier expected.

In Surfside, on Florida's east coast, workers overnight used explosives in the controlled demolition of the still-standing portion of a collapsed condo building -- a job accelerated for fear Elsa might topple the structure.

But on Monday, Miami-Dade County Mayor Daniella Levine Cava told CNN that officials were "very hopeful" that, with Elsa's current path, they would not have to pause search-and-rescue efforts after all.

- States of alarm -

The storm is expected to approach the Florida Keys -- the archipelago at the state's southern tip -- sometime Tuesday on its journey northward.

The National Hurricane Center, in an 8 pm update (0000 GMT), said Elsa had maximum sustained winds of 50 miles per hour (85 kilometers per hour) as it churned across Cuba near Havana at a speed of 13 miles per hour.

States of alarm were sounded in the provinces of Havana, Mayabeque and Artemisa as thousands were evacuated from their homes -- efforts complicated by the pandemic as Cuba endured its worst chapter of it yet.

"Protecting ourselves against Elsa cannot mean letting our guard down against Covid," Prime Minister Manuel Marrero said.

Shops closed in Havana and driving was restricted to civil defense vehicles.

The storm was expected to dump up to 15 inches (38 centimeters) of rain in parts of the island, the NHC said. 

"This will result in significant flash flooding and mudslides" it said, with Insmet warning that effects from the storm could continue in the coming days.

Elsa claimed two lives in the Dominican Republic and a third in the island state of Santa Lucia during its earlier approach through the Caribbean, the Caribbean Disaster Emergency Management Agency (CDEMA) said.

Cuban President Miguel Diaz-Canel said late Sunday on Twitter that there had only been damage to farm crops.

Elsa became the first hurricane of the Atlantic season on Friday before weakening to tropical storm status on Saturday.

Its advent represented the earliest date ever that a fifth named storm -- which does not typically arrive before August -- has struck the region.