England takes a 1-0 lead in the five-test series as James Anderson claimed two wickets and Steven Finn one on the final morning, while Australia lost six batsmen for just 66 runs after resuming on 238 for four.
"We don't want to get too far ahead of ourselves, but if we keep these standards we have a good chance of winning the series," England captain Andrew Strauss said.
Finn made the crucial breakthrough when Mike Hussey mistimed a pull to Anderson at mid-on to depart for 52.
Three wickets in four balls effectively ended Australia's chance of drawing the match. Anderson dismissed Brad Haddin caught behind for 12 and Ryan Harris lbw with consecutive deliveries.
Harris' wicket gave him two first-ball dismissals, known as a "king pair." He is only the second Australian batsman after Adam Gilchrist in 2001 who has reached this dubious milestone in test cricket.
Two balls later, Swann had Marcus North trapped lbw for 22. England's appeal was initially turned down but the video umpire ruled that the ball had struck the under-pressure Australian batsman in front of the stumps and would have gone on to dislodge the bails.
Swann, who ended with a five-wicket haul, then bowled Xavier Doherty and Peter Siddle to complete England's 100th test victory over Australia.
"We've been totally outplayed this game, there's no hiding that," said Australia captain Ricky Ponting. "Now we have to find a way to get better for (the third test in) Perth."
Before the wickets fell, North had a reprieve on 3 when Swann struck his pad, but on this occasion the video umpire ruled the batsman not out.
In the same over Prior dropped Hussey after Swann found his former Northamptonshire team-mate's edge.
Three overs later, with England having taken the new ball, the crucial breakthrough came.
Siddle also had a remarkable reprieve when the ball hit the underside of his bat and went onto hit the base of his stumps but did not remove the bails.
Kevin Pietersen was named man of the match after he scored a career-best 227 in England's first innings of 620 for five declared and took the crucial wicket of Michael Clarke with the last ball of the fourth day.
"It's pretty incredible, (England) having been so bad four years ago to be 1-0 up in Ashes," Pietersen said.
"If we keep doing the right things we'll win the Ashes."
