(CNN) -- Danielle Dayton and Patrick Murray received the call Saturday afternoon: Their 4-pound son, Brayden Murray, was shipping out.
Patrick Murray and Danielle Dayton say their tiny son's evacuation went smoothly.
As coastal hospitals prepared for Gustav, tiny babies in the Newborn Intensive Care Units were among the first to be evacuated.
"The NICU babies are our first priority when it comes to evacuation," said Keith Darcey, a spokesman for East Jefferson General Hospital in Metairie, Louisiana, where Brayden was a patient before being evacuated to Woman's Hospital in Baton Rouge, Lousiana.
Woman's Hospital started receiving phone calls Wednesday from coastal-area hospitals inquiring whether there was room in their NICU, spokeswoman Jodi Conachen said. By Sunday evening, the hospital was taking care of 32 evacuated babies.
"The hospitals started preparing early this time," she said. "They learned their lesson from Katrina."
The babies have done well since arriving in Baton Rouge, Louisiana, said Darcy Gann, director of the NICU at Woman's Hospital. "They act like they were barely affected by the transfer."
The babies arrived by ambulance without their parents. Dayton and Murray drove to Baton Rouge to join their son after dropping off their 7-year-old daughter, Breanne, to stay with relatives far from the storm. While the couple took care of their children, they had a friend board up their business in Metairie.
"You have to deal with it the best you can. You can't panic, because it doesn't do anyone any good," Murray said. "You do what's best for the baby."
Although Murray said Brayden's evacuation went smoothly, one hospital administrator said the patient evacuation process had been "incredibly frustrating." Watch part of the NICU evacuation »
"There was at least a six- to eight-hour delay in evacuating the babies," said Dr. Kevin Jordan, chief medical officer of Touro Infirmary in New Orleans, which sent seven NICU babies to Woman's Hospital. "It was very, very, very disorganized."
Jordan blamed the state office of Homeland Security and Emergency Preparedness. He said Touro is still waiting to evacuate four critically ill adult patients on ventilators. "It's been incredibly frustrating," he said.
The secretary of the Louisiana Department of Health and Hospitals said each hospital is responsible for evacuating its own patients, adding that Touro asked for assistance Sunday morning, days after other hospitals had asked for help.
"It's never a perfect process," he said. "It's never easy to evacuate thousands of people in 48 hours."
Touro, which usually has between 215 and 245 patients, has evacuated all but around 65 to 70, he added.
"The sickest of the sick patients we're keeping and sheltering in place," Jordan said. "We're concerned that evacuating them could do significant harm."
Other hospitals took a different approach. East Jefferson evacuated only its three NICU babies, keeping its 210 adult patients in the hospital.
"We survived Katrina quite well and learned a lot from the experience," Darcey said, adding that since Katrina, the exterior of the hospital has been fortified.
"But you don't want to take any chances with the little babies," he added. "Their temperatures have to be very carefully regulated, and if the air conditioning goes out, that could have a detrimental effect on their health."