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.