The expected deluge of rain failed to save the Australians as they resumed in warm sunshine 137 runs behind on 238-4.
After Michael Hussey fell for 52 in the sixth full over, James Anderson struck twice with successive deliveries.
Graeme Swann took the final wicket an hour before lunch by bowling Peter Siddle to finish with figures of 5-91.
It was an exceptional performance from England, with almost everything going according to plan from the first over of the match, when Jonathan Trott ran out Simon Katich and Ricky Ponting was caught at slip off the next ball.
There was a brief moment of concern on the final morning when wicketkeeper Matt Prior failed to cling on to a thin edge from Hussey three balls after the obdurate left-hander had recorded his 23rd Test fifty, prompting fears that the man known as 'Mr Cricket' might somehow help Australia scrape a draw.
A reprieve for such a renowned occupier of the crease, coupled with uncertainty over the weather, might have led to some anxiety for England but 10 balls later Prior was relieved to see Hussey heading back to the pavilion.
The 35-year-old Western Australian attempted a reckless, out-of-character pull at the impressive Steven Finn, who used the new ball intelligently in the absence of the injured Stuart Broad, who has been ruled out for the series.
Hussey was not in control of the shot as the ball reared up higher than he expected and he could only top edge straight to Anderson at mid-on.
England still had to contend with the combative Brad Haddin, who had made a battling half century in the first innings, and he struck successive fours off Anderson.
However, the Lancashire swing bowler soon took his revenge when an exceptional delivery moved away and caught the edge, giving Prior a routine catch behind the stumps.
The denouement as Peter Siddle is the last Australia wicket to fall
Ryan Harris offered no shot to the next delivery, which swung back the other way, and was given out lbw, a decision confirmed after a referral to the TV umpire.
He became only the second Australian, after legendary wicketkeeper-batsman Adam Gilchrist, to be dismissed for a king pair - out first ball in both innings - in a Test match.
Marcus North was the only senior batsman left and well though he played for 22, he was sent on his way after England referred an lbw appeal by Swann and replays indicated the ball would have gone on to hit the middle stump.
Australia were capitulating dramatically in much the same way England did at the hands of Glenn McGrath and Shane Warne on numerous occasions, including a final-day collapse in Adelaide four years ago.
On that day England lost nine wickets for 60 runs, a feeling Australia became increasingly familiar with as North departed to become the fourth wicket to fall in the space of six overs for the addition of only 25 runs.
Xavier Doherty's unhappy match ended when he was bowled by Swann, who fittingly ended the contest with a perfect off-break that turned in between Peter Siddle's bat and pad to give England their 100th Test victory against Australia, their first win against them by an innings for 24 years, and only their ninth in 30 Tests at Adelaide.
The final wicket also gave 31-year-old Swann, playing only his 26th Test, his 10th five-wicket haul, making him the second England spinner, after Derek Underwood with 17, to reach the mark.
For Australia, the torrential rain they had hoped for did eventually arrive late in the afternoon but by then the England celebrations were in full swing.
And, needing only to draw the five-match series in order to retain the Ashes, it is the tourists who are very much in the ascendancy as the series moves west to Perth for the third Test beginning on 16 December
After Michael Hussey fell for 52 in the sixth full over, James Anderson struck twice with successive deliveries.
Graeme Swann took the final wicket an hour before lunch by bowling Peter Siddle to finish with figures of 5-91.
It was an exceptional performance from England, with almost everything going according to plan from the first over of the match, when Jonathan Trott ran out Simon Katich and Ricky Ponting was caught at slip off the next ball.
There was a brief moment of concern on the final morning when wicketkeeper Matt Prior failed to cling on to a thin edge from Hussey three balls after the obdurate left-hander had recorded his 23rd Test fifty, prompting fears that the man known as 'Mr Cricket' might somehow help Australia scrape a draw.
A reprieve for such a renowned occupier of the crease, coupled with uncertainty over the weather, might have led to some anxiety for England but 10 balls later Prior was relieved to see Hussey heading back to the pavilion.
The 35-year-old Western Australian attempted a reckless, out-of-character pull at the impressive Steven Finn, who used the new ball intelligently in the absence of the injured Stuart Broad, who has been ruled out for the series.
Hussey was not in control of the shot as the ball reared up higher than he expected and he could only top edge straight to Anderson at mid-on.
England still had to contend with the combative Brad Haddin, who had made a battling half century in the first innings, and he struck successive fours off Anderson.
However, the Lancashire swing bowler soon took his revenge when an exceptional delivery moved away and caught the edge, giving Prior a routine catch behind the stumps.
The denouement as Peter Siddle is the last Australia wicket to fall
Ryan Harris offered no shot to the next delivery, which swung back the other way, and was given out lbw, a decision confirmed after a referral to the TV umpire.
He became only the second Australian, after legendary wicketkeeper-batsman Adam Gilchrist, to be dismissed for a king pair - out first ball in both innings - in a Test match.
Marcus North was the only senior batsman left and well though he played for 22, he was sent on his way after England referred an lbw appeal by Swann and replays indicated the ball would have gone on to hit the middle stump.
Australia were capitulating dramatically in much the same way England did at the hands of Glenn McGrath and Shane Warne on numerous occasions, including a final-day collapse in Adelaide four years ago.
On that day England lost nine wickets for 60 runs, a feeling Australia became increasingly familiar with as North departed to become the fourth wicket to fall in the space of six overs for the addition of only 25 runs.
Xavier Doherty's unhappy match ended when he was bowled by Swann, who fittingly ended the contest with a perfect off-break that turned in between Peter Siddle's bat and pad to give England their 100th Test victory against Australia, their first win against them by an innings for 24 years, and only their ninth in 30 Tests at Adelaide.
The final wicket also gave 31-year-old Swann, playing only his 26th Test, his 10th five-wicket haul, making him the second England spinner, after Derek Underwood with 17, to reach the mark.
For Australia, the torrential rain they had hoped for did eventually arrive late in the afternoon but by then the England celebrations were in full swing.
And, needing only to draw the five-match series in order to retain the Ashes, it is the tourists who are very much in the ascendancy as the series moves west to Perth for the third Test beginning on 16 December
