The Socceroos went into the second-leg of their play-off tie having secured a 1-1 away draw during the first match, but they quickly fell behind during Tuesday’s return game to a strike from Syrian forward Omar Al Somah.
However, just minutes later veteran striker Cahill levelled things up for Australia, before the 37-year-old netted his second of the game during extra time to seal the hosts their place in next month's intercontinental play-off.
After the match Postecoglou was quick to pay tribute to Cahill, declaring that the former Everton ace has to be regarded as the greatest ever Australian player.
"Timmy was Timmy and that's simply why he's the greatest ever," Postecoglou said. "He's just a freak. I'd hazard to think when the last time he played 90 minutes was, let alone 120.
"He's just got real belief in himself, a real unique and extraordinary individual and he led from the front today."
Australia will now face the fourth-place team in the final CONCACAF standings, which will be either the United States, Panama or Honduras, as the Aussies bid to secure their place at a fourth successive World Cup.