John Millman beat Brazil's Thiago Monteiro 6-7, (6), 7-6 (3) 7-6 (3) in the deciding match.
Millman took the court after Australia's doubles team of James Duckworth and John Peers lost to Brazil's Marcelo Demoliner and Felipe Meligeni Rodrigues Alves 5-7, 7-5, 7-6 (6).
On Friday, Millman came from a set and break down to beat Thiago Seyboth Wild 4-6, 7-6 (0), 6-2 to give Australia its 2-0 lead heading into Saturday's matches.
Australia was playing without its two top-ranked players. Nick Kyrgios has a left wrist injury that forced him out of a tournament in Mexico and Alex De Minaur aggravated an abdominal injury which sidelined him for last month's Australian Open.
Australia joins 11 other qualifiers and the 2019 semifinalists __ Canada, Britain, Russia and Spain __ as well as the 2020 wildcards, France and Serbia, in November in Madrid.