Jedinak's second successful spot-kick of the night in the 65th minute rescued a point for the Asian champions.
Jedinak's second successful spot-kick of the night in the 65th minute rescued a point for the Asian champions, who slip out of the automatic qualifying places in Group B.
Thailand had tried to reschedule the game because of mourning for King Bhumibol Adulaydej, who died last month, but it went ahead with fans urged to adopt a respectful tone.
Many wore black and laid candles outside the stadium in memory of the king, but as a pulsating match unfolded -- marked by three penalties -- cheers and drums rang out.
Australia started briskly and they were rewarded with a penalty on eight minutes that was stroked home by their bearded captain Jedinak, of Aston Villa.
But Thailand struck back through star striker Teerasil Dangda, who bundled the ball into the net after a quick-fire move exposed the Socceroos' defensive frailties.
The goal galvanised the home side, with Thailand playing smooth patterns through the middle towards the constantly menacing Teerasil.
And just before the hour-mark, Mathew Leckie hacked down the home side's captain Theerathon Bunmathan as he charged into the penalty box.
Teerasil swept home the penalty, sparking delirium in the stands. But Thailand's 2-1 lead was short-lived as Jedinak also went down in the box under the merest of touches from Sirod Chatthong.
Despite Thailand's protests, Jedinak converted the penalty and the visitors came closest to a winner when Nathan Burns struck the bar with a fizzer of a shot from outside the box.
While Australia drop to third, Thailand stay rooted to the foot of the table despite earning their first point in what was an impressive display.