The former England captain only lasted 10 balls before he fell to pace spearhead Mitchell Starc in the third over to get the Australians a crucial early wicket in the five-Test series.
But in-form opener Mark Stoneman and James Vince steadied the tourists to take them to 59 for one at lunch after winning the toss at the Gabba.
Stoneman, who has scored a century and three half- centuries in four innings so far on tour, was unbeaten on 25 with Vince not out 32.
Cook edged Starc to Peter Handscomb at first slip for just two. It continued an uncertain start to the Ashes tour for the Cook, who has just one half-century in five innings so far.
It was a big psychological wicket for the Australians, as Cook amassed 766 runs at 127.66 in England's 3-1 series victory in Australia in 2010/11.
Stoneman and Vince did well to keep the tourists at just one wicket with Starc, Josh Hazlewood and Pat Cummins largely contained and forcing skipper Steve Smith to bring off-spinner Nathan Lyon into the attack in the 18th over.
Lyon had no wicket for eight off six overs at lunch.
Opener David Warner (neck) and lower order batsman Shaun Marsh (back) were both passed fit to play after injury concerns.
Australia have a formidable record at Brisbane's intimidating 'Gabbatoir', where they have not lost a Test match since 1988.
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