A fortuitous deflection presented the striker with a chance that he finished with aplomb, but it was the one highlight in an otherwise underwhelming World Cup warm-up clash
The Hull City striker, perhaps in an offside position, capitalised on a fortunate deflection just before the hour-mark to score the game's only goal in Bahia.
In a major worry for Croatia coach Niko Kovac, left-back Danijel Pranjic came off in the first half after twisting his ankle under a challenge.
Croatia started with regulars Ivan Rakitic, Ivica Olic, Vedran Corluka and Mario Mandzukic – suspended for their opener against Brazil – on the bench.
Mandzukic came on for Jelavic just after the goal and the Bayern Munich striker required treatment on his ankle late on.
The absence of Rakitic in particular hurt Croatia, who, like their opponents, lacked the cutting edge to threaten often.
Australia were compact throughout, but never appeared dangerous in the attacking third as they look ahead to their World Cup opener against Chile on June 13.
Ange Postecoglou tested the central defensive pairing of Matt Spiranovic and Alex Wilkinson, while regular winger Tommy Oar played as the number 10.
Captain Mile Jedinak returned from an ankle injury, while Mark Bresciano came on in the 66th minute.
Dario Vidosic was given a start ahead of James Troisi and Ben Halloran in perhaps the biggest surprise.
And the Socceroos enjoyed the better of the early moments as they pushed Croatia deep, albeit without seriously threatening.
The European side created the half's only real chance in the 20th minute as Eduardo, having been floored and requiring treatment moments earlier, forced Mat Ryan into a save down low to his right after a Nikica Jelavic cutback.
Both teams looked settled in possession, but a lack of creativity meant there were few clear-cut chances in a cagey first half during which hopeful long passes and crosses were relied upon.
Croatia goalkeeper Stipe Pletikosa was involved for the first time 90 seconds after the interval, easily claiming a Tim Cahill header.
Darijo Srna came close with a free kick in the 56th minute, but he curled a powerful 20-yard effort wide of Ryan's goal.
Just two minutes later and Croatia were ahead through Jelavic. Luka Modric's run and pass deflected off Australia defender Ivan Franjic and into the path of Jelavic, who took his time before striking past Ryan and in off the underside of the crossbar.
The striker appeared to be offside when Modric's pass was played but the goal stood.
Second-half substitute Ivica Olic tested Ryan from range 15 minutes later as, with the earlier introduction of Rakitic, Kovac's men started to look more dangerous.
Croatia held firm in defence, just as they will need to do on Thursday in the World Cup opener against hosts Brazil, while Australia need to work on their attacking combinations ahead of their meeting with Chile a day later.