The storm brought 150mph (240km/h) winds when it made landfall and those people who did not flee have been advised to shelter in place.
One person was killed when a tree fell on their home in Ascension Parish, in the Baton Rouge area.
Ida will test New Orleans’ flood defences, strengthened after Hurricane Katrina killed 1,800 people in 2005.
President Joe Biden said Ida would be “life-threatening”, with immense devastation likely beyond the coasts.
Over 750,000 homes in Louisiana are without power, and Mr Biden said it could take weeks to restore supplies.
Ida gathered strength over the warm waters of the Gulf of Mexico during the weekend.
It made landfall on Sunday south of New Orleans as a category four hurricane – meaning it will cause severe damage to buildings, trees and power lines. It has since weakened slightly to a category three storm.
In some places the storm surge could be as high as 16ft (4.8m), potentially submerging parts of the low-lying coastline.
Ida came ashore on the 16th anniversary of Katrina, a category three storm.
Since then, billions of dollars have been spent on flood defences, known as levees. So far, the levees have held, though a flash flood warning is in place for New Orleans.
“There is no doubt that the coming days and weeks are going to be extremely difficult for our state and many, many people are going to be tested in ways that we can only imagine today. But I can also tell you that as a state we’ve never been more prepared,” state Governor John Bel Edwards said.
High winds tore part of the roof off a hospital in the town of Cut Off, Louisiana, just inland from the Gulf of Mexico. The hospital said it had suffered “significant damage” but that its patients were safe.
The impact of climate change on the frequency of storms is still unclear, but increased sea surface temperatures warm the air above, making more energy available to drive hurricanes.
As a result, they are likely to be more intense with more extreme rainfall.
New Orleans resident Tanya Gulliver Garcia, who works for the Center of Disaster Philanthropy, told the BBC she was staying put.
“I’m fairly worried – more than I thought I would be. I’ve volunteered in disasters for a number of years… but it’s different when it’s in your own home, instead of a nice, safe, sheltered place,” she said.
In a tweet, the US National Weather Service (NWS) told New Orleans residents: “Go to an interior room or a small room with no windows. Stay put during this time.”
Louisiana hospitals are already under pressure from Covid-19. The state has the third-highest rate of infections in the US.
Normally, hospitals in the predicted path of the hurricane would be evacuated, but this time there are few beds available, even at facilities further inland.
“We don’t have any place to bring those patients. Not in state, not out of state,” Mr Edwards said.
More than 90% of oil production in the Gulf of Mexico has been shut down. (Courtesy BBC)