Australia bounced back from Wednesday’s Davis Cup defeat by Great Britain to come from behind and beat France 2-1.
But Alex de Minaur beat Ugo Humbert 7-6 (7-2) 6-3 before Purcell and Matt Ebden won the deciding doubles rubber 7-5 6-3 for last year’s runners-up.
That kept alive Australia’s hopes of qualifying from Group B for the quarter-finals in Malaga in November.
Great Britain face Switzerland in their second group match on Friday, with qualification for the last eight set to go down to the hosts’ final game with France on Sunday.
Defending champions Canada cruised past Sweden for a second straight Group A win in Bologna, despite being without Felix Auger-Aliassime, Milos Raonic and Denis Shapovalov.
Meanwhile, the United States suffered their first defeat in Group D, losing to the Netherlands in Split.
The Czech Republic backed up their Group C win over Spain by defeating South Korea in Valencia, but world number one Novak Djokovic could return to action for Serbia against hosts Spain on Friday.