Their hard yards, however, counted for nothing when open-side flanker Sam Warburton was eventually isolated and penalised for holding on.
In what would become a defining feature of the match, Australia were bullied in the next scrum - Faingaa penalised for popping out under the force of Wales' front row. Jones, though, misfired off the tee, hooking his kick past the left-hand post.
In an apparent attempt to profit from Shane Williams' lack of height, Quade Cooper launched a dangerous cross-field kick towards the right corner.
However, his punt was marginally overcooked and O'Connor could only knock forward on the touchline as he tried to keep the ball alive.
Australia were again battered in the scrum before Warburton turned over again. From the resulting foray forward, Mark Chisholm was blown up for pulling Alun Wyn Jones' arm in a line-out, and Jones sent over his second penalty to make it a one-point game.
O'Connor had the chance to restore his side's advantage when Tom Shanklin high-tackled Genia, but the Western Force star's 40m kick fell short and wide of the posts.
Australia laid siege to the Welsh line again but Faingaa burrowed over in vain as the referee ruled the scrum-half had made it to the line courtesy of a double-movement.
Jones squandered another penalty just before the half-time whistle that would have had Wales out in front after Cooper was penalised for a high tackle on Shane Williams.
The home side were fortunate to escape when Matt Giteau uncharacteristically fumbled as Australia worked the ball menacingly wide after the restart.
But, moments later, fortune favoured the Aussies when Beale's kick bounced off Bradley Davies and into the hands of Genia, who released O'Connor on the right.
The 20-year-old marauded up the wing, brushing off a feeble tackle from Shane Williams before releasing an inside pass to Beale, who ran over unchallenged, O'Connor converting.
Australia's pitiful scrummaging yielded another penalty for Wales, which Jones landed just outside the 22 to make it 9-14.
Beale almost scored a sensational try when he collected his own chip before grubbering past Hook only to spill the ball as he tumbled towards the line.
The Wallabies pulled 10 points clear, though, on the hour. Cooper robbed Bradley Davies, and when the ball was eventually flung wide to O'Connor he released Alexander who dived over for his first Test try.
O'Connor was wayward with his touch-line conversion attempt before the television match official ruled that Alun Wyn Jones' lunge for the line had fallen short.
Rees went over for Wales late on
O'Connor sent over another penalty to give the visitors a commanding 22-9 lead.
Wales centre Shanklin was sin-binned for an early tackle as Pocock drove for the line, but the hosts never gave up and they were given a lifeline with nine minutes remaining.
Australia conceded penalties at two scrums five metres from their own line, and from the third, Wales number eight Jonathan Thomas fed Rees - on for Mike Phillips - to dive over following a long period of pressure.
Replacement Dan Biggar landed the conversion, but another O'Connor penalty with six minutes left killed off Wales' late resurgence.
Wales: Hook, Harries, Shanklin (Czekaj 76), Bishop, Shane Williams, Stephen Jones (Biggar 65), Phillips (Richie Rees 65), Jenkins (James 71), Matthew Rees (Bennett 75), Adam Jones, Davies, Alun Wyn Jones (Deiniol Jones 74), Lydiate, Warburton (Martyn Williams 65), Thomas.
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