After the visitors claimed both opening-day singles matches at Melbourne's John Cain Arena, new British doubles pairing Jodie Burrage and Harriet Dart completed a rout in the best-of-five qualifying tie on Saturday.
Their 6-3 6-4 win over Storm Hunter and Ellen Perez gave Great Britain an unassailable lead, sending last year's semi-finalists back to September's eight-nation finals in China.
Australian rising star Emerson Jones restored a hint of pride for the home team with a 7-5 6-3 win over Katie Swan in their dead-rubber singles match.
But the 3-1 result means Australia miss the finals for the second straight year and will return to the playoffs in November to fight for a spot in next year's qualifiers.
"We're pretty disappointed right now," Stosur said.
"It certainly wasn't the result we'd hoped for and envisioned having, but it wasn't through lack of trying or preparation.
"We were ready to go and our players have been in pretty good form leading in, but unfortunately that's the way the dice rolls sometimes and we've got to learn from this.
"We'll definitely be back in November bigger and stronger, and try to do well next time."
Despite injured world No.30 Maya Joint's withdrawal, Australia had gone into the qualifying tie as strong favourites against a Great Britain team missing their best four players.
But individual rankings counted for little as Dart led the British charge.
The world No.173 upset Kimberly Birrell 4-6 6-3 6-3 in their singles clash on Friday, building on 17-year-old debutant Mika Stojsavljevic's shock 7-6 (7-4) 7-5 win over Talia Gibson.
Big-serving Stojsavljevic arrived in Melbourne ranked 219 places below world No.56 Gibson, who had knocked off five top-20 opponents in a sparkling run of form in the lead-up.
Facing a 2-0 deficit, Australia had the weight of history against them on Saturday.
Only nine teams have recovered from that position to win since the best-of-five format was introduced in the BJK Cup - formerly known as the Federation Cup - in 1995.
"On paper the Aussies were favourites. Even though (Friday) was amazing, being 2-0 up, I don't think we could leave anything to chance today," Great Britain captain Anne Keothavong said.
"The players were given a game plan that they were able to execute and they were able to perform, but in my head it was kind of preparing for that fifth rubber.
"You've just got to be ready for all eventualities and anything else is a bonus. I'm just so proud of all the players for what they were able to do in difficult circumstances."
One-time "orange girl" Jones, 17, relished her BJK Cup debut on home soil as she overcame Swan.
"It was really fun and such a great experience, and I'm lucky to have been given that," the former junior world No.1 said.
"I just had fun out on court and tried to get us a game, so I'm really happy with how it went."
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