Britain led 2-0 overnight at the inaugural mixed international team competition after wins for Cameron Norrie and Katie Swan, meaning they needed one victory from the three matches on Friday to clinch the win.
That arrived in the first match as Dart defeated Maddison Inglis 6-4 6-4.
Dart, the British number one in the absence of Emma Raducanu, who chose to skip the event, would have expected a rematch against Australian number one Ajla Tomljanovic, who she defeated at the Billie Jean King Cup last month.
But Tomljanovic was a late withdrawal with a knee injury and Dart made no mistake against 180th-ranked Inglis.
"Maddy played really great tennis today and I had to work for it," Dart said on unitedcup.com. "First match of the year, I was really nervous, but the crowd were great."
Australia came out on top in the final two rubbers, with Jason Kubler repeating his Wimbledon upset of Dan Evans.
Evans looked certain to force a decider when he led 5-0 in the second set but Kubler staged an impressive fightback to win 6-3 7-6 (3).
John Peers and Sam Stosur then clinched a 7-6 (4) 6-4 victory over Dart and Jonny O'Mara in the mixed doubles.
Britain can clinch a spot in the semi-finals with victory against Spain over the weekend but they go in as underdogs given Spain boast three top-20 players in Rafael Nadal, Pablo Carreno Busta and Paula Badosa.
In the other ties there were victories for Greece, USA, Italy, France and Switzerland.