Cook, who carried his bat through the England first innings in the fourth Melbourne Test for an unbeaten 244, looked on his way to another big score until Josh Hazlewood struck in the 33rd over.
Hazlewood's leg before wicket appeal was turned down on- field, but on review the ball was found to be just pitching in line with the left-hander's leg stump and hitting middle and Cook was given out for 39 off 104 balls.
At tea, England were 122 for three after winning the toss with skipper Joe Root and Dawid Malan both having reached 16 not out.
Fellow opener Mark Stoneman earlier hit some nice drives among his four fours before he nicked a rising Pat Cummins delivery to Tim Paine for 24 off 24 balls in the 10th over.
James Vince looked set after 82 minutes at the crease before he shaped for a square cut at a wide Cummins delivery only to be caught behind for 25 in the 29th over.
After a rain delay and an early lunch, play finally got under way at 12.40pm (0140 GMT).
Seamer Chris Woakes was ruled out with a side strain, opening the way for young leg-spinner Mason Crane to make his debut for England.
Pace spearhead Mitchell Starc returned to Australia's bowling line-up at the expense of Jackson Bird after missing the drawn fourth Melbourne Test with a bruised heel.
Australia have already clinched the Ashes with an unassailable 3-0 lead in the five-Test series.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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