About 80,000 lightning strikes were recorded, some damaging power generation facilities, while winds toppled over 22 transmission towers.
Power was restored to Adelaide area by 03:00 local time (17:30 GMT) but tens of thousands of homes remain cut off.
On Thursday, authorities advised people to leave work early as winds were expected to intensify again.
How 'unprecedented' storm played out
The state premier said "twin tornadoes" had hit northern areas in "an extreme weather event" on Wednesday. Jay Weatherill compared the storm to Hurricane Sandy which hit New York in 2012.
He said officials in charges of the power grid had told him "any system would not be able to cope with a weather system of this kind".
"We had winds which were so strong that when they hit power lines they created such energy they were tearing the towers out of the ground."
While South Australia's wild weather was widely forecast, no-one could really predict the impact it would have on everyday life.
In Adelaide, commuters were left feeling their way in the dark as the lights went off on their way home. Gridlock followed, as trains and trams came to a standstill.
At the airport, some passengers had to depend on handwritten boarding passes, after power for the printers failed. Some describe the whole experience as "apocalyptic", while one woman, who was on the operating table in the local hospital and about to undergo surgery, said it was simply "funny".
What has been predictable is that an extreme weather event has very quickly turned into an Australian political storm, with the state's dependency on renewable energy now being debated with the full force and bluster of a tornado.
The Bureau of Meteorology (BOM) said the storm was a once-in-50-years event.
It is forecasting further heavy rainfall throughout Thursday and into Friday. Further gales of up to 140km/h (87mph) are expected, and flood warnings remain in place.
On Thursday, South Australia's State Emergency Service has warned residents to prepare for more high winds and rain, cautioning there was "a potentially very dangerous" weather system gathering, with winds of up to 125km/h (78mph) predicted.
"As the winds increase, rainfall intensity is also expected to increase," said Chief Officer Chris Beattie.
"High tides this afternoon and evening will result in significant storm surges" he added, recommending that workers go home early.
Influential South Australian Senator Nick Xenophon has called for a independent inquiry.
"This is a disgrace. How did this happen? How is an entire state blacked out?" he said.
The Australian Energy Market Operator, which controls the nation's electricity network, said it was still working to understand the cause of the fault.
It urged Australians to "take care during this outage and to follow the safety advice and guidelines issued by the South Australian government and emergency services".

The remains, believed to be those of a 71-year-old woman, were found in Fenny Bridges, Honiton at about 16:00 GMT on Wednesday, police said.
A 66-year-old man is being held in custody and questioned at Heavitree Road police station.
The woman has yet to be formally identified but a post-mortem will be carried out later, police said.
Devon and Cornwall Police said it was not looking for anyone else in connection with the incident.
South Australia is enduring a second day of severe weather, after the entire state was left without power overnight.
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