They all initially tested negative shortly after arrival but two became symptomatic and were tested on Sunday, with the results coming back positive on Monday morning.
The flight has not been listed as an exposure site as the three family members tested negative two days later and all other passengers remain in isolation as fellow red zone returnees.
Authorities are also on high alert after two members of a three-person removalist crew who travelled through Victoria and SA from NSW tested positive for the virus.
Removalists are permitted under the border permit system.
The crew made stops at two Victorian family homes on 8 July, one in City of Whittlesea and the other in Maribyrnong.
The City of Whittlesea family had recently relocated from interstate and tested negative on 9 July.
"They had very limited exposure out in the community," Mr Weimar told reporters on Monday.
"If they are positive, it will be a short window. We know their movements."
The second family of four from Maribyrnong are staying in temporary accommodation while awaiting a longer-term move. It is likely they will be shifted into hotel quarantine.
Contact tracers are working to establish the movements of the first removalist who returned a positive result, though it is believed he travelled from Sydney to Melbourne via the Hume Highway and worked at several homes in the city before travelling to SA.
NSW Health alerted Victoria's authorities late on Sunday night.
Speaking at 3pm on Monday, 30 minutes after receiving an update on test results, Mr Weimar said another of the three removalists is also believed to have "turned positive as well".
All three removalists have since returned to Sydney.
Mr Weimar believes the crew stayed within their cab but said the crew's initial stories were not quite stacking up.
"It is taking us a little bit longer than we would like. We are also talking to their employer who is providing fuel cards and logbooks and various other bits and pieces," Mr Weimar said.
The cases come after Victoria recorded its 12th day of no locally acquired cases and the state effectively shut the border to NSW and the ACT overnight, declaring them red zones under the travel permit system.
ACT Chief Minister Andrew Barr said it was "incredibly disappointing and frustrating" Canberra was classified the same as Sydney, despite recording no community COVID-19 cases for more than a year.
But Victorian Health Minister Martin Foley said the rationale behind the decision was "simple", noting the ACT is surrounded by regional NSW.
Victoria Police Acting Deputy Commissioner Michael Grainger said an extra 90 officers had been sent to the border, on top of the 260 operating there since last week.
Police are continuing to keep border travellers guessing, adopting roving patrols, pop-up checkpoints and the air wing division scanning from Mildura to Mallacoota.
Mr Grainger said only six fines have been issued out of the 28,000 people stopped at the NSW border so far.
Victorians in NSW are still allowed to enter the state but now must isolate at home for 14 days.
In addition, the Victoria-NSW "border bubble" arrangement will remain for local residents, though they must carry proof of address and stay within the bubble.