Overall, Australia won a World Cup game for only the third time in their history.
Duke scored midway through the first half with a glancing header past keeper Aymen Dahmen to give Australia the lead.
The victory snapped Australia’s seven-match winless run at World Cups since they last picked up three points in a victory over Serbia in 2010.
Australia sit second in Group D with three points from two games, having been well-beaten 4-1 by holders France in their tournament-opener.
Australia’s best World Cup performance came in 2006 when they also reached the last 16, in the days of Tim Cahill, Harry Kewell and Mark Viduka.
(With agency inputs)