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 arrived at Adelaide Oval on the fifth day knowing that hope springs eternal, or at least until Joe Root gets out.
"We always had the faith in the team and the belief", Australia skipper Steve Smith said.
0 - When Root and Woakes both failed to add to their overnight scores, England were on the fast track to defeat.
The teams now move to Western Australia where England will have to rally to stay alive in the series. "We thought if we could come out and bowl really well, hit our lengths and get one or two wickets then things can happen pretty quickly".
"Disappointing. I don't think we did ourselves justice", Root said.
Hazlewood, otherwise below his best in the second Test, nearly hit the 150 km/h mark in a fired-up opening spell that also featured the dismissal of nightwatchman Chris Woakes.
The edge Hazlewood found was so slight a bemused Woakes sent the decision to the DRS, and though Hot Spot showed no mark, Snicko was unequivocal and the nightwatchman was removed. Off-spinner Nathan Lyon packed off Moeen Ali and left-arm pacer Mitchell Starc then took the last three wickets to wrap up a win which ended a roller-coaster Test match.
Australia were now into the England tail and Craig Overton (7) departed plumb lbw to the first delivery with the new pink ball, an inswinger from Starc.
For the record, Starc bagged Stuart Broad caught-behind and Bairstow was last out chopping on to his former Yorkshire team-mate.
The game for England resumed at 178 with six wickets in hand but the wickets collapsed like a house of cards and Australia registered their second consecutive victory in just two hours of the game to dominate the Ashes. 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.
Australia dismissed England for 227 on Monday but decided against sending them back in to bat despite holding a 215-run first innings lead.
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