JAMESTOWN, Australia—Tesla Inc. Chief Executive Elon Musk may have overpromised on production of the company’s latest electric car, but he is delivering on his audacious Australian battery bet.

An enormous Tesla-built battery system—storing electricity from a new wind farm and capable of supplying 30,000 homes for more than an hour—will be powered up over the coming days, the government of South Australia state said Thursday. Final tests are set to be followed by a street party that Mr. Musk, founder of both Tesla and rocket...