A scrambled injury time reply from Alberth Elis was no concern for Australia. Postecolgou became the first Australian to steer his country to a World Cup, even if doubts linger as to whether he'll take the team there.
On Wednesday night, Honduras set out to stymie Australia, with a five-man defence and go-slow orders from coach Jorge Luis Pinto from the first whistle.
Postecoglou struck a frustrated figure on the bench as Australia set about unpicking the Honduran lock.
Interplay between Aaron Mooy and Tom Rogic looked promising, but it took 37 minutes before the Socceroos first threatened.
Tim Cahill's trickery allowed Aziz Behich to cut back for Rogic, only for the Celtic man to mis-time his shot.
The lack of action quietened the crowd, growing fears.
But after the break, Australia found a decisive moment.
The Hondurans couldn't lay a glove on Rogic, with flicks and turns that set him apart, and on 53 minutes he was cut down at the top of the box.
From the free kick, Jedinak stepped up and fired past wrong-footed Honduran goalkeeper Denis Escobar - with officials crediting the goal to Henry Figueroa owing to his deflection. Needing a goal, Honduras suddenly became interested in possession.
But the Socceroos were on top and continued to create chances.
Cahill hit the bar from a floating header as Australia set up camp in their forward line. The pressure told, with Bryan Acosta handling the ball to bring Jedinak to the spot.
The skipper sent his spot kick within inches of the post with an unstoppable effort on 72 minutes.
He sent Escobar the other way when substitute Robbie Kruse was scythed down by Johnny Palacios in the box, ensuring a golden night for Australia.
