Premier Steven Marshall confirmed the first case was a miner who worked at the Northern Territory mine linked to an outbreak and had previously tested negative but has since returned a positive test at home in Adelaide.
His wife and three of his four children have also tested positive.
The entire family has now been moved to a dedicated quarantine facility.
Despite the new cases, Mr Marshall said a lockdown was not happening.
“I will start this press conference by saying South Australia will not go into lockdown,” he said.
“I think many people will be extraordinarily relieved by that.”
The miner developed symptoms after returning a negative test while in isolation.
None of the positive cases recorded today spent time in the community while infectious, the premier said.
Mr Marshall said the family’s caution made South Australia “very fortunate”.
“This family has been doing the right thing, all of the miners who have returned from the Northern Territory have been doing the right things by isolating and getting tested, we have not had any other positive tests for the other 28 miners,” he said.
The five latest cases are South Australia’s first COVID-19 cases in 200 days.
SA’s Chief Health Officer Professor Nicola Spurrier said the miner is a man in his 30s.
His wife is also in her 30s. The three children who tested positive are all aged under 10.
The family also has a baby who Professor Spurrier said “was not yet positive”.
“What we see with this variant is that there is almost 100 per cent hit rate in infections,” Professor Spurrier said.
Despite ruling out a widespread lockdown, Mr Marshall said new restrictions will apply from today, but urged people against unnecessary panic buying.
When out in public, South Australians are urged to wear a mask, private gatherings are now limited to a maximum of ten people and people should work from home if they can.
The premier implored South Australians not to rush to the shops to panic buy, but admitted that “frankly, it is a natural human behaviour”.
“Obviously people are concerned with the media speculation this morning, but there is no lockdown. Can I say that with Australia we have excellent supply of all food and other appropriate groceries,” Mr Marshall said.
“Frankly, it is a natural human behaviour, but again I will just say to people, please be sensible about this, no lockdown in South Australia, no need whatsoever for panic buying.”
South Australia is already under mild COVID-19 restrictions.
Masks are mandatory in all high-risk settings such as aged care homes and hospitals.
Weddings and funerals are capped at 150 people for the week.
Pubs and clubs are restricted to one patron for every two square metres of floor space.
The restrictions were introduced at midnight last night and are due to last for a period of one week.