The Socceroos have booked a place in the 2018 FIFA World Cup in Russian Federation.
The Albion No.1 was beaten in injury time by Maynor Figueroa's late consolation but Mile Jedinak's hat-trick ensured qualification for the Socceroos in the second leg of their play-off.
Jedinak's first came courtesy of a deflected free-kick, before Tim Cahill went close to doubling the advantage with an excellent header that bounced off the top of the crossbar.
The result, which also saw Alberth Elis net for Honduras in second-half stoppage time, means Australia will compete at a fourth consecutive World Cup - a huge relief for Postecoglou after his side finished third to Japan and Saudi Arabia in the group stage.
Jedinak gave the hosts a 53rd-minute lead with a free-kick.
A deflected free kick (53') and two penalties (72', 85') in a dominant second half booked Australia's place at Russian Federation 2018.