It was a third stalemate in succession for the Socceroos, who dropped to third in Asian Confederation Group B as Japan beat leaders Saudi Arabia 2-1.
A win over a Thailand side managed by former Town player Kiatisuk ‘Zico’ Senamung would have put Australia top at the halfway stage.
The Socceroos are in a six-nation group with home and away games running through to next September.
The top two in each of the two groups qualify for the 2018 finals.
The two third-placed sides will play each other home and away to determine which takes part in an inter-federation play-off against a nation from CONCACAF (North, Central American and Caribbean).
Mooy, who is on loan from Manchester City, played for the full 90 minutes.
He will now return to join in preparations for Town’s Championship clash at Cardiff City on Saturday.
Australia needed two penalties, both scored by Aston Villa’s Mile Jedinak, to gain a hard-earned point.
Zico’s Thailand had lost all four of their previous qualifiers.