Peru substitute Edison Flores headed against the foot of Australia's post early in the second period of extra time but that was as close as both teams came to breaking the deadlock as the game drifted towards a shootout.
Pedro Gallese saved Martin Boyle's opening spot kick. Australia drew level when Luis Advincula hit the post.
Redmayne, making only his third appearance for the national team and dancing along the goal-line, saved from Valera, and it was enough for the Socceroos to return to Qatar in November for their sixth overall World Cup appearance.
"This idea was floated pre-selection that this (shootout) might eventuate in these kinds of circumstances," Redmayne said. "At the end of the day, it's the flip of a coin. Either right or left."
Australia will meet defending champion France in the opening game of Group D on Nov. 22, and take on Tunisia and Denmark.
The final place at the World Cup will be filled in the same stadium tomorrow when Costa Rica meets New Zealand.