The win is just Australia's second from their past eight games and gives Graham Arnold's team something to build upon heading into Tuesday's must-win clash with UAE.
Aaron Mooy was given his first competitive action since a 2-2 draw with Oman in February in a new-look starting XI as Martin Boyle, Mitch Duke, Joel King, Mathew Leckie and Adam Taggart were all rested.
But Australia found themselves behind in the 18th minute when Al-Taamari's long-range shot from well outside the box after some fine build-up play gave Maty Ryan no chance.
The Socceroos were lucky to escape further damage later in the first half when Wright caught Jordanian forward Ali Olwan inside the penalty area, but the referee waved play on.
Mooy, making his 50th appearance for Australia, nearly broke the deadlock early in the second half when his free kick crashed into the post, with Nicholas D'Agostino unable to direct a follow-up effort on target.
The former Premier League midfielder, who has been reduced to training in Scotland with the Chinese Super League suspended due to that country's COVID-19 situation, would eventually be substituted after a valuable 61 minutes.
Shortly after Mooy's withdrawal, Australia took the lead.
Jordan goalkeeper Yazeed Abulaila saved smartly from another Wright header, this time from a corner, with Mabil able to acrobatically finish after D'Agostino headed the loose ball back towards goal.
Australia will now turn their attentions to the clash with UAE, knowing a victory in that game will up another playoff against South America's Peru in Qatar the following week for a spot in the World Cup.