The tourists, who ended an eight-game losing streak against their Australian counterparts by reducing the series lead to 3-1 on Friday, lost the toss and home captain Michael Clarke decided to bat.
Stuart Broad took two wickets to dismiss the hosts' opening two batsmen, with Aaron Finch (7) bowled out before Shane Watson (0) followed his partner into the pavilion after slashing at a wide and full one, with Jos Buttler on hand to make the catch.
It got worse for the Aussies as Clarke, who was rested in the defeat in the fourth ODI, departed after just eight runs as Tim Bresnan disturbed the stumps, leaving the hosts 43-3 after just 14 overs.
The New Zealand-born right-hander bowled to Shaun Marsh (36), whose attempt to flick one through the side leg ended up in the clutches of Cook, who made amends for an earlier drop.
Stokes shortly claimed England's fifth and sixth wickets of the contest, bowling out Glenn Maxwell (22), whose outside edge was caught by Buttler, before ending the resistance of George Bailey (56) at the crease by dismissing the right-hander shortly after completing his half-century.
Broad followed suit by picking up his third wicket, bowling out Matthew Wade with a full delivery after the left-hander showed too much of the stumps to his opposite number.
Chris Jordan brought an end to England's innings in style, bowling a full toss for James Faulkner (27), who struck straight into the hands of Eoin Morgan, before having the chance to bowl for his hat-trick after Nathan Coulter-Nile (15) was dismissed lbw to leave the hosts 217-9 at the end of play.