Troisi broke Korean hearts with the very last kick in the first extra period after Son Heung-Min had snatched a magical injury-time equaliser at the end of normal time.
Australia's triumph gave them a first title since defecting from Oceania nine years ago, while South Korea's anguish continued after they failed to break a hoodoo in the tournament stretching back to 1960.
"It was a super effort from everyone," said Australia coach Ange Postecoglou. "It was a final, it never goes to script. It was tough, you've got to grind it out and the courage the players showed tonight was enormous."
"We pride ourselves on being very fit and it was just a matter of taking our chance. I'm super-proud of them. Hopefully from now on, us Aussies can take on the world because I have a real belief in these guys."
Australia's historical triumph vindicated Postecoglou's decision to overhaul the team after its "golden generation" passed their prime.
It could also further antagonise countries said to be behind a movement to oust Australia from the Asian confederation after shock revelations in the build-up to the final.
But South Korea's superstar Son produced a moment of sheer brilliance in injury time, bursting through to fire past goalkeeper Mathew Ryan after a clever flick from captain Ki Sung-Yueng.
Australia, the Asian Cup's top scorers with 14 goals, were forced to dig deep in a reversal of the group meeting between the two sides earlier in the tournament, which was dominated by Australia but won 1-0 by the Koreans.
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