The Socceroos had been held to a goalless draw in the first leg in San Pedro Sula last week but had a trio of second-half strikes from the skipper in Sydney to thank for securing their ticket to Russia.
Jedinak's first came courtesy of a deflected free-kick, before Tim Cahill went close to doubling the advantage with an excellent header that bounced off the top of the crossbar.
But Cahill's close call would not prove costly for Ange Postecoglou's men, who soon left Honduras needing two goals as Jedinak punished a Bryan Acosta handball by firing home from the penalty spot.
