Australia made 294 for nine after being sent in to bat but they had been in danger of not batting out their overs when they slipped to 192 for six in the 29th over.
All-rounder Andile Phehlukwayo, 20, took four for 44 in his second one-day international, taking three top-order wickets in a frenetic first half of the Australian innings.
Bailey (74) and Hastings (51) steadied the innings, with Hastings going to a maiden international half-century before he became Phehlukwayo's fourth victim when he was caught on the long-off boundary.
Phehlukwayo had Finch caught at backward square leg and Australian captain Steve Smith leg before wicket in his second over. He followed up with the wicket of Mitchell Marsh, caught by a diving wicketkeeper Quinton de Kock.
With David Warner setting the tone with 40 off 36 balls, Australia scored rapidly but kept losing wickets before Bailey and Hastings came together.
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