Ricky Ponting's team looked to have no hope of winning the match after England piled on 551-6 in their first innings and still led by 107 with nine wickets in hand early on the last day.
But the wheels suddenly fell off on the last morning when master leg-spinner Shane Warne ripped through the English batting lineup and the tourists collapsed for 129.
Australia then raced to their victory target of 168 with 19 balls to spare.
''I've played in a lot of good wins before but this beats the lot. It's one of the all-time great Test wins as far as I'm concerned,'' Ponting told a news conference.
''To turn a Test match around like that, it just doesn't happen.
We've all seen enough games of Test match cricket and I've played enough to know that you have to do something exceptional.
''Our cricket over the last three days has been as good Test match cricket as you'll ever see from any team.''
Heavily Criticised: Australia's win gave them a 2-0 lead in the series with three matches to go after their 277-run win in Brisbane last week but Ponting was coy about making predictions for the remaining matches.
Australia won the first Test in England last year but lost the series 2-1 to concede the Ashes and the skipper was reluctant to say this series was already as good as over.
''We haven't even thought about that yet,'' said Ponting, who was named man of the match after scoring 142 and 49.
''We're all very happy and excited with what we've just achieved over the last three days of the game.'' But Ponting, who was heavily criticised for some of his decisions in the last Ashes series, said the manner of England's defeat would hurt the tourists after they were in such a strong position.
''There's no doubt about that. They wouldn't have even thought that there was a chance that they could lose that,'' Ponting said.
''They probably would have been thinking about getting themselves into a position where they could chuck us into bat late today and take a few wickets and hopefully put us under pressure.
''But that wasn't the way it was. We had them under pressure all day and the game was there for the taking.''