Redmayne danced across the goal line before diving to the right to block a kick from Alex Valera and give Australia a deserved win at Ahmed Ben Ali Stadium.
Redmayne was brought in within three minutes of the match for a penalty shootout and became an instant hero with his antics on the line as he looked to distract the opposing player.
He danced up and down, shaking his hips and throwing his arms in a throwback to the clown antics of Liverpool’s Bruce Grobbelaar when his side won the European Cup in 1984.
Australia will play in Group D of the World Cup with defending champions France, Denmark and Tunisia. The finals run from November 21 to December 18.
Australia missed the first penalty but turned the next five to silence thousands of Peru fans, who traveled to the match and gave a raucous support but saw their side conjuring few chances.
Instead, a player like Australia dominated the early exchanges, with winger Martin Boyle twice cutting through the Peruvian defense to deliver exciting crosses.
The second half followed a pattern similar to the first with limited efforts on goal until the 80th minute when Australia suddenly had three good chances to win the match.
Peru captain Pedro Galeese saved the free kick, which was easily carried out by Agden Hrostic, and after five minutes, Aziz Bahish broke two steps and suddenly found himself in front of the goal, but he narrowly missed as he tried to straighten his shot into the net.
Then running to the left of Australian substitute Auer Mabel saw him slip the ball in Hrustic’s way but he didn’t get enough power in his shot and Galeese saved again.
After nine minutes of extra time, Australian Matthew Ryan was finally tested with a stinging Edison Flores shot that was saved by the goalkeeper.
Flores then headed toward the post as the South American team found extra strength in the closing stages but were still unable to decide the score after two hours of play, paving the way for Redmayne to emerge as champions.
The play-off between the fifth-placed Asian qualifiers and South America placed 31st in this year’s World Cup finals.
On Tuesday, last place will be decided in the finals when Costa Rica and New Zealand meet in the intercontinental play-off, as well as at Ahmed Ben Ali Stadium.