Redmayne danced across the goal-line before diving to the right to stop a kick from Alex Valera and hand Australia a deserved victory at the Ahmad bin Ali Stadium.
Redmayne had been brought on three minutes from the end of the game for the shootout and became an instant hero with his antics on the line as he looked to distract the rival kicker.
He danced up and down, wiggling his hips and throwing his arms about in a throwback to the clowning antics of Liverpool’s Bruce Grobbelaar when his team won the European Cup in 1984.
Australia will play in Group D at the World Cup with holders France, Denmark and Tunisia. The finals run from November 21 to December 18.
Australia had missed the first penalty but converted the next five to silence thousands of Peru supporters, who traveled for the game and provided noisy support but saw their team conjure up a few opportunities.
Instead, a workmanlike Australia dominated the early exchanges, with winger Martin Boyle twice cutting through the Peruvian defence to provide teasing crosses.
The second half followed a similar pattern to the first with limited efforts on goal until the 80th minute when Australia suddenly had three good opportunities to win the match.
Ajdin Hrustic’s tame free kick was easily saved by Peru captain Pedro Gallese and five minutes later Aziz Behich broke through two tackles and suddenly found himself in front of goal but narrowly missed as he tried to curl his shot into the net.
Then a run down the left by Australia substitute Awer Mabil saw him slip the ball into the path of Hrustic but he did not get enough power on his shot and Gallese saved again.
Nine minutes into extra time Australia’s Mathew Ryan was finally tested by a stinging shot from Edison Flores that the goalkeeper saved.
Flores then headed against the post as the South Americans found an extra gear in the closing stages but they were still unable to settle the result after two hours of play, setting the stage for Redmayne to emerge as a hero.
The single game playoff between the fifth-placed finishers in Asian and South American qualifying determined the 31st place at this year’s World Cup.
On Tuesday, the last place at the finals will be decided when Costa Rica and New Zealand clash in their inter-continental playoff, also at the Ahmad bin Ali Stadium.