They got off to the perfect start as Andre Carrillo volleyed home Paolo Guerrero's cross, beating an outstretched Mathew Ryan, for La Blanquirroja's first goal at a finals since 1982.
The result saw the South American's leapfrog Australia into third spot.
The Socceroos needed to beat Peru and hope France did them a favour against Denmark to stand any chance of reaching the knockout stages, but a slow start proved too tough a mountain to climb.
Australia continue their run of never holding a lead at half time in a World Cup game, this being the nation's 16th match in tournament history, and the task got significantly harder shortly after the restart.
Guerrero turned goalscorer, on what is expected to be his World Cup swansong, as he disguised a half-volley behind Trent Sainsbury after 50 minutes.
Tim Cahill was then introduced, himself also playing his final World Cup, but could not inspire a comeback from Australia.
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