Puerto Rico Governor Pedro Pierluisi said more than 1,000 stranded residents were rescued after Hurricane Fiona hit the US territory on Monday.
The storm knocked out the island's power supply and caused catastrophic flooding and landslides after landfall, officials said, leading to 30 rescue operations. One man died while operating a generator.
The storm ripped asphalt from roads, swept away a major road bridge, swamped cars, forced airports to close and dumped so much rain that some rivers rose up to six metres in a few hours, witnesses said.
"This has been catastrophic," Mr Pierluisi said.
US President Joe Biden called him on Monday, promising federal support in the weeks to come as people are advised to stay home or in shelters with rescues continuing.
Mr Pierluisi said more than 2,000 people were in more than 100 shelters on the island. Only 30 per cent of the island has running water, he said.
Fiona, which hit five years after Hurricane Maria devastated the Caribbean island, was moving towards the Dominican Republic on Monday.
"The President said that he will ensure that the Federal team remains on the job to get it done, especially given that Puerto Rico is still recovering from the damage of Hurricane Maria five years ago this week," the White House said.
New York also said it was sending more than 100 Spanish-speaking State Police officers to the island.
The centre of the storm made landfall on the south-western coast of Puerto Rico near Punta Tocon at 3.20pm, with maximum sustained winds of about 140kph, making it a Category 1 hurricane, the National Hurricane Centre said.
Luma Energy, operator of the island's grid, said the entire electrical system had been shut down to protect its infrastructure.
Some power was being restored on Sunday night, with priority given to hospitals and essential community services, but reconnecting the whole island would take days.
Puerto Rico's grid remains fragile after Hurricane Maria knocked out 80 per cent of power lines in September 2017. The Category 5 storm killed more than 3,000 people on the island of 3.3 million.
No deaths were reported from Fiona by Sunday night, but authorities said it was too early to estimate the damage as the storm was forecast to cause torrential rain across Puerto Rico on Monday.
Officials said Fiona caused several landslides, while a bridge in the central town of Utuado was washed away by a swollen river. Ports were closed and flights out of the main airport cancelled.
US President Joe Biden approved an emergency declaration for Puerto Rico on Sunday, authorising the Federal Emergency Management Agency to co-ordinate disaster relief and provide emergency protection.
A wide expanse of Puerto Rico was forecast to receive between 30 and 40 centimetres of rain, while some areas could be hit by up to 63.5cm, the NHC said.
Authorities opened more than 100 shelters and closed beaches and casinos.
Torrential rains and mudslides are also forecast for the Dominican Republic as the storm moves north-west, with the Turks and Caicos Islands expected to be affected on Tuesday.
Aid agencies in the Dominican Republic began evacuating high-risk areas in the east of the country on Sunday.
President Luis Abinader postponed a trip to New York for the UN General Assembly, while the start of the Dominican school year was pushed back from Monday to Wednesday.
One death tied to Fiona has been reported so far, in the French Caribbean island of Guadeloupe.
Authorities said a man was found dead on Saturday after his house was swept away by floods. France will recognise a state of natural disaster for Guadeloupe, President Emmanuel Macron said.
— With reporting from agencies.