Inglis was a late call-up for the match after Ajla Tomljanovic withdrew with a knee injury.
Dart’s 6-4, 6-4 win on Friday meant Britain were victorious with two matches to spare.
Dan Evans lost to Jason Kubler 6-3 7-6 (7-3) and Dart and Jonny O’Mara were defeated 7-6 (7-4), 6-4 by Sam Stosur and John Peers in a doubles match.
Great Britain will now prepare to face their other group opponents Spain, with the first match being contested between Norrie and Rafael Nadal on Saturday.
Swan will take on Nuria Parrizas-Diaz on the same day.
The inaugural mixed-team tournament involves six round-robin groups of three teams.
Great Britain, Spain and Australia are competing for a chance to play either the United States, Germany or the Czech Republic in a play-off to reach the semi-finals.
Meanwhile Nick Kyrgios has accused Australia co-captain Lleyton Hewitt of “throwing him under the bus”.
Kyrgios withdrew from the United Cup mixed teams tournament in order to be in good shape for the Australian Open and his co-captain’s public response frustrated the 27-year-old.
Hewitt said he was unaware of Kyrgios’ withdrawal and was frustrated at the lack of communication.
After Tomljanovic withdrew from her match against Dart, Kyrgios tweeted to question whether she would receive the same treatment.
“Mmm I wonder if Sam Stosur [Australia’s co-captain] will throw her under the bus like our captain did for me… ‘hard to prepare when you don’t know what’s going on’,” he wrote.