John Manenti's side went into the ninth and final tournament in second place, two points behind South Africa, and in reach of an historic double after the Australian women won their World Series in May.
Their hopes were boosted when Commonwealth Games gold medallists South Africa were shock losers to the United States and New Zealand to bomb out before the quarter-finals.
Australia, Argentina and Fiji now all remain in with a chance to be crowned Series champions come Sunday evening (local time).
Australia blitzed the pool stage, beating Japan 61-0, Argentina 19-14 and Spain 54-0.
Against fellow title contenders Argentina, the South Americans took a 14-5 lead at halftime through converted tries from German Schulz and Matías Osadczuk while Henry Hutchison did the business for Australia.
Dietrich Roache and Maurice Longbottom then scored second half tries - with two conversions from the former - for Australia to storm back to take an important win.
The men in gold have never won a World Series title, with their best finish a runner-up placing back in 2000-01.
"The Aussie men have never won the World Rugby Sevens Series - so this is a rare occasion for the squad," Manenti said.
"You never know how many chances you will get in your career to achieve something like this, so everyone is determined to really go out there and take the opportunity.
"On the back of the women's success in winning the World Series, the men are motivated and keen to emulate their performance and make it a rare double for any Sevens national teams."
Australia will take on Kenya in the first Cup quarter-final on Sunday before Olympic champions Fiji meet Team USA.
Samoa will play Argentina and New Zealand, who trumped their pool, line up against Ireland.
The tournament semi-finals and final also will be played on Sunday (local time).
Due to the affect of COVID-19 on the Sevens calender, final placings will be decided on teams' best seven results in the nine rounds.
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