The third test starts December 14 in Perth, where England is in danger of surrendering the urn after winning the Ashes on home soil in 2015.
The paceman delivered a hefty blow when he removed Chris Woakes with the second ball of the day and two overs later landed a knockout punch when he had Joe Root caught behind before the England captain could add to his overnight 67.
"We always had the faith in the team and the belief", Australia skipper Steve Smith said. "Unfortunately losing the two early wickets really hampered our chances", Root said. "I don't think we did ourselves justice today".
"The way we went about things in the second innings proved we are still massively in this series - we have shown we can outperform Australia but we haven't done it for five days".
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Sometimes we have had the odd disagreement over decisions but what matters is that we are both playing for the same team. I'd urge everyone not to forget what this team has achieved. "I can't change what people think, I'm not bothered by it".
Contact was so feint, in fact, that Aleem Dar's decision was upheld after Woakes went to DRS only on the basis of a Snicko spike and in the absence of any corroboration at all from Hotspot technology.
That left England lurching towards defeat on 188 for seven after 45 minutes of play on the final day.
England had therefore lost four wickets for 19 runs, dating back to Dawid Malan's dismissal late on Tuesday night.
Starc needed only one delivery with the second new ball to dismiss the stubborn Craig Overton, also LBW and this time knowing no review was required.
For the record, Starc bagged Stuart Broad caught-behind and Bairstow was last out chopping on to his former Yorkshire team-mate. It didn't work then, and it failed this time, too, with Shaun Marsh's unbeaten 126 - he was named the man of the match - helping Australia post 442-8 declared in the first innings.
The highest winning fourth innings at the Adelaide Oval remains 315 for six by Australia against England in 1902.