The heaviest rain -- almost 16 inches (400 millimeters) in just a few hours, beating historical records for the entire month of April -- hit provincial La Plata overnight. A day earlier, the capital of Buenos Aires was hit hardest.
About four more inches (100 millimeters more) of rain were expected before the bad weather passes on Thursday, the national weather service said.
At least 2,500 people were evacuated from their homes to about 20 centers in the La Plata area, which is about 37 miles (60 kilometers) southeast of Argentina's capital.
The flooding threatened to ruin food supplies across La Plata's metropolitan area, which has nearly 1 million people.
It also closed the private Spanish Hospital, a complex that covers an entire city block, after waters rushed into the basement, cutting power and destroying X-ray machines and other diagnostic equipment.
"We're sending away all the patients and the hospital will be closed for several days," said Sebastian Sambron, one of the hospital's top officials. "We're telephonically cut off, and without power since last night. The hospital is collapsed."
National Planning Minister Julio de Vido estimated that 280,000 people remained without power across the city and surrounding province of Buenos Aires, where most Argentines live.
"Our job is focused on restoring service, but we're going to wait until the equipment dries to guarantee the safety of the electricity workers, because we don't want any deaths," De Vido said.
YPF said no injuries were caused by the refinery fire, which it blamed on "an extraordinary accumulation of rainwater and power outages in the entire refinery complex." The impact on Argentina's chronically short fuel supplies wasn't immediately clear.
The six killed in Buenos Aires included a subway worker who was electrocuted and an elderly woman who drowned inside her home. Many evacuees slept in their cars overnight, and still had standing water in basements, parking lots and storage rooms.
The governments of Fernandez and Buenos Aires Mayor Mauricio Macri sought to blame each other for the chaos, and the nation's divided media focused their coverage in ways that put one side or the other in the worst light.
Macri said Wednesday that the only solution is for the constantly warring governments to work together on expensive and long-term public works projects, creating huge underground drainage pipes to carry increasingly common torrential rains out to the Rio de la Plata.
"Facing the magnitude of what we've lived through, I insist that public works are what will change this story," Macri said, describing one such project that was achieved through regional cooperation and a World Bank loan -- the kind of borrowing that Fernandez has sought to avoid.
"We need to do the same with all the waterworks that are needed in the city, in greater Buenos Aires and in the province of Buenos Aires," Macri said.
Associated Press Writer Cristian Kovadloff in La Plata and Almudena Calatrava, Debora Rey and Michael Warren in Buenos Aires contributed to this story.