Not since 1958 had there been so few points in a match between these sides, but the win boosts Wales' confidence ahead of next year’s Rugby World Cup.
Australia lock Ned Hanigan was penalised for not rolling away after making a tackle and Biggar kicked Wales to what will be a famous win, in a contest that saw both sides commit numerous unforced errors and show a lack of cutting edge in attack.
Wales’s Leigh Halfpenny, usually so deadly off the tee, landed two penalties but also missed two very kickable three-pointers, the second from directly in front of the posts on a nervous night in Cardiff.
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Leigh Halfpenny converted two penalties against the Wallabies on Saturday
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It looked as though Wales' wait for a win over the Wallabies would be extended after Bernard Foley’s 75th minute penalty was greeted by great boos under the roof.
But three minutes later, Biggar kicked over the winner to put this hoodoo to bed.
It was by no means a classic, but for Wales coach Warren Gatland it is a potentially pivotal moment in his build-up to the global showpiece in Japan next year when the sides will meet again in the first-round pool stage.
Losing seven of 10 Tests this year means Australia's search for a morale-booster ahead of facing England on November 24 rolls on to facing Italy in Rome next Saturday.
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