Police said it was an Islamist-inspired plot, but they did not link the plan to a specific terrorist group.
Officers became aware people in Sydney were allegedly planning to carry out a terrorist attack using an "improvised device," Australian Federal Police Commissioner Andrew Colvin said.
Turnbull described it as an elaborate conspiracy that involved bringing down an airplane.
The suspects were rounded up in raids in four Sydney suburbs, federal police and the New South Wales Police Force said in a news release. They are in custody but have not been charged, Colvin said.
Police did not specify the date or location of the threat to Australia's aviation industry.
"Exactly what is behind this is something that we will need to investigate fully," Colvin said.
A house in the suburb of Surry Hills, where one of the suspects was arrested, remained cordoned off Sunday, with two police officers stationed outside.
More police personnel were parked in a nearby alley and investigators appeared to be examining the property.
Neighbors told CNN that a family, including at least three children, lived at the house. They said the family had been friendly enough, though they did not know them well.
Husnain Muhammad, 24, said he was worried about a backlash against Muslims in Australia after Saturday's arrests. "Everyone will see us as terrorists and we're not," he said.
Police said searches were also carried out in the Sydney suburbs of Lakemba, Wiley Park and Punchbowl.
Extra security measures
The Prime Minister said extra security measures have been in place at Sydney Airport since Thursday and have since been put in place at other major airports.
Transport security officials advised travelers to get to their airports two hours before their scheduled flight departures.
"Those traveling should go about their business with confidence," Turnbull added.
Australia's terrorism threat level remains in the middle at "probable," the Prime Minister said, between "possible" and "expected."
"The number one priority of my government, and my commitment to the Australian people, is to keep them safe," he said.
Last month, ISIS claimed responsibility for an attack at a suburban Melbourne apartment building where one man was killed and three police officers were wounded.
The attacker, who was known to police and out on parole, was shot dead after he ended a standoff by bursting out of an apartment.