But Florida appeared to be getting a bit of a break, as forecasters shifted its likely path westward, suggesting not the direct hit earlier expected but more a glancing blow to the southwest coast of the state.
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. The job was accelerated for fear Elsa might topple the structure in uncontrolled fashion.
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.
The storm is expected to approach the Florida Keys -- the archipelago at the state's southern tip -- sometime Tuesday on its march northward.
The National Hurricane Center, in a 8:00 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.
It is expected to gain strength after it emerges off the coast of western Cuba Monday evening.
Cuba's meteorological institute Insmet reported earlier that Elsa had cut through the island with winds of up to 100 kilometers an hour, with "some stronger gusts."
States of alarm were sounded in the provinces of Havana, Mayabeque and Artemisa.
More than 100,000 people around the country were evacuated from coastal or low-lying areas. This was complicated by the pandemic, with Cuba enduring 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 Cuba, the NHC said. "This will result in significant flash flooding and mudslides," it said.
During its earlier approach through the Caribbean, Elsa claimed two lives in the Dominican Republic and a third in the island state of Santa Lucia, 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.
On Friday, Elsa became the first hurricane of the Atlantic season before weakening to tropical storm status on Saturday.
Elsa's advent represented the earliest ever that a fifth named storm has struck the region. Typically, the fifth named storm does not arrive before August.