The Socceroos got off to a shaky start with Massimo Luongo robbed of possession in the 10th minute which allowed Jordan to break and force Mat Ryan into an early save.
There was nothing the Australian keeper could do in the 26th minute as Anas Bani Yaseen emphatically headed Jordan in front after a well worked corner routine.
Jordan nearly made it 2-0 just minutes later with Ryan doing well to force a fierce free-kick from Abdel Rahman onto the crossbar.
A poor first-half from the Socceroos ended in frustration as they were denied what appeared to be a clear handball in the Jordan box just before the break.
Australia coach Graham Arnold replaced right-back Josh Risdon with Rhyan Grant during the interval as he looked to find a response for the Socceroos.
Jordan however weren't about to roll over as they sat back and denied the Socceroos meaningful possession of the ball.
Only the woodwork denied Awer Mabil an equaliser in the 78th minute as he did well to create space and rifle off a shot, while in the 86th minute Jamie Maclaren did find the back of the net only for the offside flag to deny him his first Socceroos goal.
Australia, who forced a fine save with the final kick of the game, ultimately made to pay for a particularly poor first-half as an inspired Jordan held on for a 1-0 win that leaves them in pole position to top the group.