Australia had the better opportunities in a 0-0 draw in the first leg last Friday in San Pedro Sula.
But after a frustrating first half, when Tom Rogic created the only real chance with a mistimed shot in the 37th, Australia dominated the second half.
Australia had opportunities to qualify directly for Russia in Asian qualifying but missed out on goal difference to Saudi Arabia in a tight finish in that group.
That forced the Australians into an Asian playoff against Syria, which veteran Tim Cahill sealed with two goals in Sydney last month to set Australia up for the last-ditch home-and-away series against Honduras.
The Australians edged Uruguay on penalties in a playoff in 2005 to qualify for the following year’s World Cup in Germany, and then qualified directly for 2010 and 2014.
Australia coach Ange Postecoglou said his Socceroos “did it the hard way, but we deserve to be there.”
The final word went to the jubilant Jedinak, a veteran midfielder who plays for second-tier Aston Villa in England.
“We got the job done and that’s all that matters — it’s going to be a big 2018,” he said.