The Bati led 16-12 in the second half and were in the hunt for the greatest upset in World Cup history but a Tom Trbojevic hat-trick after several acts of Tedesco genius lifted the Kangaroos out of trouble.
Fiji had lost all seven Tests against Australia before this one, with their closest losing margin a 34-2 defeat in a 2013 World Cup pool match.
This was different, much different, in front of 41,108 fans at Suncorp Stadium on Sunday night.
After a stellar season for the Dolphins, Fiji five-eighth Kurt Donoghoe did everything in his power to conjure the greatest win in his country's history with three try assists for Bati winger Semi Valemei to score a hat-trick.
Donoghoe's halves partner Brandon Wakeham was also up to his ears in several good things the Bati did.
Australia have qualified for the semi-finals in top spot with the win while Fiji must beat New Zealand on the Gold Coast on Saturday to join them in the knockout rounds. The Kangaroos will play at Allianz Stadium on Sunday, November 8 in their semi-final.
The Bati dominated Australia up front in a stunning display of forward punch and impact in the opening half, despite trailing 12-10 at the break.
Fiji lock Caleb Navale had a blinder. The Manly powerhouse suffered an ACL rupture in the pre-season and returned to play post-surgery in 186 days in a tribute to his commitment.
Navale was the best forward on the park in a brilliant display for his country.
The class of fullback Tedesco came to the fore when he went into first receiver and found a scintillating pass to put right centre Kotoni Staggs over to score the opening try.
Donoghoe threw a sensational cutout pass to send Valemei in before Wakeham's slick grubber was swooped upon by second-rower Taane Milne to give Fiji a stunning lead.
Tedesco's dart from dummy-half on the last tackle set up a long-range try, with half Sam Walker handling twice on the way to scoring.
Donoghoe was at it again after the break with a magical cutout ball again for Valemei to give Fiji a 16-12 lead.
An obstruction, correctly ruled, denied Solomone Saukuru a try from another Donoghoe pass.
Minutes later Tedesco found a pass for Trbojevic to score out wide after excellent lead-up work from Walker.
A Tedesco grubber ricocheted into the arms of Trbojevic for his second try and when he crossed for a third the result appeared secure.
Donoghoe had other ideas and set up Valemei again before Jacob Preston sealed the deal for Australia..
Australia fielded four debutants in half Walker, hooker Blayke Brailey, centre Bradman Best and second-rower David Fifita.
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