Centuries to opener Sam Konstas (109 off 144 balls) and wicketkeeper Josh Philippe (123no off just 87), plus half-centuries to Campbell Kellaway (88 off 97), Cooper Connolly (70 off 84) and Liam Scott (82 off 122), propelled the tourists to a massive first-innings total of 6-532 declared on the second day of the four-day match at Lucknow's Ekana Cricket Stadium.
However, when it came their turn at the crease, India A showed the Aussies they had some battling talent of their own, cruising to 4-403 by stumps on Thursday's third day.
Openers Abhimanyu Easwaran (44 off 58) and Narayan Jagadeesan (64 off 113) got the home side off to a great start with an 88-run stand.
No.3 Sai Sudharsan followed up the openers' good work with a patient 73 off 124 deliveries, before the tourists struck back.
Wickets to paceman Xavier Bartlett and allrounder Scott, backing up his impressive half-century, were followed by successes for spinners Connolly and Corey Rocchiccioli, to have the home side 4-222.
Connolly trapped Sudharsan in front for 73, before Rocchiccioli claimed the prized scalp of 14-Test player Shreyas Iyer, lbw for just eight off 13 deliveries.
That was as good as it got for the Australia A bowlers, Devdutt Padikkal (a patient 86no off 178) and Dhruv Jurel (113no off 132) steering the home side to 4-403 at stumps, trailing the tourists by 129 runs with one day remaining.
Most of the attention surrounding the Australia A tour of India involves Konstas, who remains in the reckoning to open the batting in this summer's Ashes, despite failing to fire in Australia's 3-0 Test series win over the West Indies.
Adding even more interest to the selection question is the fact that Marnus Labuschagne and Kurtis Patterson, who both boast Test experience, each scored centuries in One-Day Cup matches this week.
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