Great Britain must beat France in Manchester on Sunday to reach the Final Eight of the Davis Cup.
Leon Smith's British side would have advanced on Saturday without playing if Switzerland had beaten Australia.
However, Australia won 3-0 as Thanasi Kokkinakis and Alex de Minaur gained singles victories over Dominic Stricker and Marc-Andrea Huesler respectively.
Matthew Ebden and Max Purcell then beat Stricker and Huesler 6-2 6-4 in doubles to secure Australia's qualification.
The Australian team, runners-up in 2022, would have been facing elimination if they had lost 3-0 to Switzerland, but Kokkinakis beat Stricker 6-3 7-5 before De Minaur triumphed 6-4 6-3 against Huesler.
Ebden and Purcell, who won the men's doubles competition at Wimbledon in 2022, then produced a dominate display to take Australia to the Davis Cup knockout stages, which will be played in Malaga, Spain between 21-26 November.
'We need a full house and a lot of noise'
Smith's Great Britain beat Australia 2-1 on Wednesday and Switzerland 2-1 on Friday, and face France in their final Group B match on Sunday, with a crowd of more than 13,000 expected - a record for a single day of Davis Cup action in the United Kingdom.
"It's a massive match. It will go down to the wire - we need a full house and a lot of noise," said Smith.
A loss would mean Britain, who last won the Davis Cup in 2015, miss out on the closing stages of the tournament.
Every one of Smith's five-man squad has featured in Manchester this week.
Jack Draper, 21, won on his Davis Cup debut with a singles victory over Kokkinakis, before Dan Evans defeated world number 12 De Minaur.
Evans and Neal Skupski have twice been paired together in doubles, losing to Australia's Ebden and Purcell before beating the Swiss duo of Stan Wawrinka and Stricker.
Andy Murray, a member of the winning British side eight years ago, gained a come-from-behind win over Switzerland's Leandro Riedi, while British number one Cameron Norrie has only played once, losing to Wawrinka.
Smith acknowledged he had some tough decisions to make over his selections.
"Everyone's played now. We've got a full deck of cards to play with," said Smith. "Ultimately we'll go with what we think's the best option, but everyone's available, everybody's ready to go.
"You're not always going to get the decisions right, and there's not a right answer most of the time. You can look at match-ups as much as you want but there's other equations in there.
"Genuinely it feels everyone's in it together, everyone's been totally accepting of any decisions I've made. There's not been any ill feeling at all.
"There's always going to be disappointment because these guys are great players, they all want to play."
France have a mixture of youth and experience with 19-year-old Arthur Fils, an ATP Tour winner at the Lyon Open in May, alongside doubles specialists Edouard Roger-Vasselin, 39, and Nicolas Mahut, 41.
Adrian Mannarino and Ugo Humbert, 34th and 36th respectively in the world rankings, are also in the France squad.
The match between Great Britain and France will be shown live on the BBC iPlayer and BBC Sport website from 13:00 BST on Sunday.
Holders Canada book Final Eight spot, but United States are eliminated
Elsewhere in the Davis Cup on Saturday, Canada - top of Group A with two wins from two - secured their place in the Final Eight as Alexis Galarneau won the opening singles match with a 6-3 7-6 (7-5) triumph over Alejandro Tabilo of Chile.
In Group C, the Czech Republic clinched top spot with their third victory as they beat already-qualified Serbia in Valencia, Spain.
Eighteen-year-old Jakub Mensik beat Dusan Lusovic 6-3 6-2 and Jiri Lehecka won 7-6 (9-7) 7-5 against Laslo Djere of Serbia.
Finland moved into the Final Eight for the first time with a Group D win that eliminated 32-time champions the United States.
The Europeans took a dramatic opener as Otto Virtanen saved two match points in his 7-6 (7-5) 1-6 7-6 (9-7) victory over Mackenzie McDonald, before Emil Ruusuvuori defeated world number 13 Tommy Paul 7-6 (7-1) 6-4.
On Sunday, Italy need to beat Sweden in Bologna in Group A to have a chance of qualifying, already-eliminated South Korea and Spain play in Valencia in Group C, and the already-qualified Netherlands take on bottom-placed Croatia in Group D in Split.
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