A dozen Australians were in action at Melbourne Park on Thursday, with Queensland-based Sweeny the first through to the final round.
He was joined there by four others - former child prodigy Destanee Aiava, 17-year-old Maya Joint, Melbourne local Omar Jasika and Hunter.
The doubles world No.1 overcame Switzerland's Celine Naef 6-3 7-6 (7-4).
John Millman, who has announced his impending retirement, plays later in the day.
Wildcard Sweeny survived a tough first set against Italy's Luca Nardi and then raced away to secure a 7-6 (7-2) 6-2 victory over the 13th seed.
Another wildcard, Melisa Ercan, was set to follow Sweeny on to court but withdrew from her clash with China's Wei Sinjia after injuring her calf during her first-round match.
The teenager's consolation for her heroic opening win is the biggest cheque of her fledgling career.
Ercan will hobble away from Melbourne Park with a $44,100 pay day - more than doubling her previous career earnings.
View this post on Instagram A post shared by Tennis Australia (@tennisaustralia)
Sweeny, who is ranked No.257, is hopeful of banking another victory on Friday after he reached the same stage in 2021 but fell at the final hurdle.
"It'd be amazing. I'm at a spot now where I really think I can do it," he said after downing the world No.119.
"Not just because I'm in the third round, but I've proven myself, particularly the end of last year, that I do have the level and capabilities to beat these guys."
Sweeny said people often commented on his uncanny resemblance to former world No.1 Hewitt, and as a kid he idolised the Australian great.
"I would love to be as good as him - I respect the way he played and have idolised him, but I do want to play how Dane Sweeny plays, and I need to focus on bringing my own game-style to the court."
Hunter had to overcame a misfiring serve, with six double faults, to down Naef.
She trailed 1-3 in the second set and managed a crucial break to level at 5-5 as Naef served for the set.
Naef, the 30th seed, had plenty of chances but could only convert four of 18 break points.
"I didn't panic ... I just used my experience to work my way back. I'm glad I got it done in straight sets," Hunter said.
"My parents are flying in Sunday so I have to win again so they can see me play singles as well as doubles."
Aiava showed some grit in her clash with Switzerland's Jill Teichmann, rallying to beat the 20th seed 2-6 7-5 6-1, while American-born Joint was too good for Latvia's Darja Semenistaja 6-1 7-6 (7-4).
Wildcard Jasika survived a tough three-setter against American Zachary Svajda 7-5 4-6 6-2.
Perth teen Talia Gibson toiled hard against 16-year-old Brenda Fruhvirtova before the Czech 10th seed prevailed 1-6 6-3 7-5.
Sydneysider Ivana Popovic fell to 11th-seeded Hungarian Anna Bondar 6-4 6-4 while Edward Winter was ousted by India's Sumit Nagal 6-3 6-2.
