Holders England now face the prospect of losing the series in next week's third Test at Perth's WACA Ground where they have not won a Test since 1978.
England, resuming the final day on 176/4, were bundled out for 233 inside the first session at the Adelaide Oval.
The hosts had appeared a little rattled for much of the penultimate day in this second Test, while England captain Joe Root was sustaining England's unlikely fightback with a determined half-century.
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Josh Hazlewood struck twin blows by accounting for night watchman Chris Woakes (5) and England captain Joe Root (67).
"We've won the game - I would have thought about it had we lost", Smith said. We're still massively in the series.
After the match, it was down to BT Sports pundits Michael Vaughan, Geoffrey Boycott, and Ricky Ponting to pick apart the second Test, and it was the Australian who raised a brilliant point during his analysis.
"That has to be the benchmark how we approach the rest of the series".
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Umpire Aleem Dar had taken an age to raise his finger and rule that Woakes had been caught behind and despite no sign of an edge on Hot Spot a slight spike on Snicko convinced television official Marais Erasmus to confirm the decision.
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.
Starc had Stuart Broad caught behind for eight and finished off the Test just before tea bowling Jonny Bairstow for 36.
The result means that Australia has a ideal record in day-night Tests, having prevailed in all three of the pink-ball matches they have played in Adelaide, as well as the Test they won against Pakistan in Brisbane last summer.
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Australia dismissed England for 227 on Monday but decided against sending them back in to bat despite holding a 215-run first innings lead. That England even fought back into this match, that they entered the final day with any sort of hope whatsoever, was a surprise in itself.