The veteran midfielder scored just 17 minutes after being introduced from the bench, heading home Tommy Oar's 69th-minute corner with the help of a notable deflection at Sydney's Allianz Stadium.
It allowed Postecoglou, who replaced Holger Osieck after a 6-0 loss to France, to mark the occasion of his first match in charge with a promising victory.
Mark Milligan and Robbie Kruse starred for the home side, while youngster Tom Rogic looked dangerous in a 25-minute second-half cameo.
Australia almost suffered a disastrous start when Jason Davidson prodded the ball past his own goalkeeper following a mix up, but Ivan Franjic stepped in to clear to safety.
The hosts settled in to the game after that, with Patrick Pemberton becoming increasingly occupied in the Costa Rica goal.
After Milligan had set up Mathew Leckie to shoot wide, Kruse's heavy touch let him down after he had been sent clear by Rhys Williams' excellent throughball.
Postecoglou's men continued to probe, earning a corner from which Mile Jedinak fired a shot on goal from inside the six-yard box, but Pemberton made a comfortable save.
The first chance of the second half fell to Costa Rica but Wellington Phoenix striker Kenny Cunningham miscued with only goalkeeper Mat Ryan to beat.
Cahill was brought on in the 52nd minute, replacing Leckie, and he rose highest to nod in Oar's excellent delivery, with the ball taking a wicked deflection off Junior Diaz on its way in.