Tornado warnings were also issued from Louisiana to Florida as the storm's powerful winds knocked down power lines and blocked roads with fallen trees.
Oil companies evacuated workers from offshore rigs ahead of the arrival of Lee, a disorganised but major rainmaker.
Lee came ashore just 80km southwest of Lafayette, packing sustained winds of 75km/h, the Miami-based National Hurricane Center said.
The slow-moving but massive storm was expected to continue to draw moisture from the Gulf as it gradually drifted north to drench the Appalachian mountains and Tennessee River valley.
With some areas forecast to receive up to 50cm of rain over the Labor Day holiday weekend, officials warned residents of coastal states as well as landlocked Kentucky and Tennessee to prepare themselves for extensive flooding.
"A lot of it has been flash flooding, where the water's rising quite quickly," said Corey Pieper, a meteorologist with the National Weather Service.
"Roadways is where it gets really dangerous because people think they can make it across," he said.
Too often, the floodwaters are hiding the fact that the road has been washed out, he said. And even if it's still there, the flood can be far more powerful than most people expect.
"It only takes a few inches of water to wash even a truck away," Pieper said.
Lee was battering the Gulf Coast six years after the region was devastated by Hurricane Katrina.
The levee system around New Orleans failed after Katrina, putting much of the city underwater. More than 1500 people died.
On Monday last week, Katrina's sixth anniversary, the Times-Picayune newspaper reported that an upcoming Army Corps of Engineers report gives the levee system a "near-failing grade", despite a US$10 billion post-Katrina rebuilding job.
The intense rain that Lee was dumping on the city is expected to provide the most severe test of the levee and canal systems at Lake Pontchartrain and elsewhere since Hurricane Gustav came close to overwhelming the levees three years ago.
Earlier this week New Orleans mayor Mitch Landrieu said the water pumps that are key to the city's flood mitigation are "100 per cent operational".
By Saturday afternoon all 24 pumps were operating at full capacity and one station was forced to briefly switch to back-up generators due to a temporary power failure.
Most residents were taking the storm in stride.
"After going through Katrina, there's always that 'what if?' in the back of your mind," said Liz Wellman, 28.
"But I would say 90 per cent of me felt completely safe because we were dealing with a tropical storm, and it's nothing compared to what Katrina was."
US forecasters are also monitoring Tropical Storm Katia in the Atlantic Ocean.
Forecast models vary, and Katia is still well out to sea, but some tracks show the storm nearing and perhaps clipping the US eastern seaboard sometime next week.
-AAP
