England were 13-6 down early in the second half and would have been 10 points adrift had Quade Cooper kicked a penalty.
But two tries in seven minutes from captain Chris Robshaw and fly-half Owen Farrell turned the tide as England regained the Cook Cup against a Wallaby side they will face in the pool phase of the 2015 World Cup in England.
Defeat extended Australia’s poor 2013 record to eight defeats from 11 matches, with just three wins.
“You’ve got to give it to the English,” Australia captain Ben Mowen told Sky Sports. “They scrambled really well and had more urgency.”
England, who scored 14 unanswered second half points, were without five injured British and Irish Lions in centres Manu Tuilagi and Brad Barritt, Tom Croft, lineout specialist Geoff Parling and prop Alex Corbisiero.
England coach Stuart Lancaster gave a Test debut to Joel Tomkins, the centre becoming a dual code international, while the pack saw brothers Mako and Billy Vunipola start a Test together for the first time.
“We had a lot of young lads making their first starts at Twickenham. I think we deserved it in the end.”
Australia coach Ewen McKenzie sprang a surprise earlier in the week when he stripped James Horwill of the captaincy and installed Mowen as his new skipper, with mercurial fly-half Cooper the vice-captain.
England’s pack, playing under the new scrum engagement laws for the first time, forced repeated penalties at the set-piece.
But Farrell was unable to take full advantage in a first half where he landed just two out of five penalty attempts.
Australia made England pay with the opening try in the 31st minute.
Cooper’s excellent wide pass found Israel Folua and the full-back skipped round England wing Chris Ashton’s attempted tackle before he was held up short of the line.
However, the ball was recycled and eventually Matt Toomua burst through opposing centre Billy Twelvetrees’s tackle from a couple of yards out for a converted try.
Cooper then kicked another penalty to give Australia a 13-6 half-time lead.
The England side that beat Australia in the 2003 World Cup final gave the home crowd something to cheer with a half-time parade around the pitch.
But it was Australia who pressed early after the break, with Marland Yarde’s late tackle on opposing wing Adam Ashley-Cooper giving the Wallabies a penalty only for Cooper to miss for the first time in the match.
England full-back Mike Brown, the man-of-the-match, counter-attacked with a clever step from near his own goal-line, although replays suggested he had a foot on the touch-line, meaning the ball ought to have been called ‘dead’.
However, the ball was worked to Yarde but he was tackled into touch by Ashley-Cooper.
From the ensuing lineout close to Australia’s line, it seemed England lock Courtney Lawes had knocked-on.
Play continued, however, with Wallaby scrum-half Will Genia’s clearing kick charged down by Mako Vunipola and Robshaw pouncing on the loose ball for a 50th minute try to the delight of a capacity crowd of over 80,000.
Farrell converted and the match was level at 13-13.
And Farrell’s day got even better when he burst through a gap between No 8 Mowen and hooker Stephen Moore for England’s second try.
The officials checked for a possible obstruction by England replacement hooker Dylan Hartley on Moore but decided it was insufficient to disallow the score.
Farrell, who scored 15 points in all, converted and England led 20-13.
Cooper missed a long-range penalty with 15 minutes left and, although Australia pounded away from a close range lineout, England’s defence held firm.
England face Argentina at Twickenham next weekend when Australia play Italy in Turin.
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