At tea on the third day, England, following-on, were 123 for three in their second innings -- still 209 runs behind Australia's first innings 481, which featured captain-elect Steven Smith's 143.
Left-handed opener Cook was 54 not out and Jonny Bairstow 20 not out.
England at least had the consolation of having won the Ashes at 3-1 up in the five-match series.
Opener Adam Lyth, desperate for a big score to cement his England place, saw his miserable Ashes continue when, on 10, he edged a good length ball from Peter Siddle that cut away off the pitch to Australia captain Michael Clarke at second slip.
His slow trudge off the ground suggested Lyth knew his Test future was hanging by a thread.
Ian Bell, on nine, saw a pull off Mitchell Marsh sting the fingers of Chris Rogers at mid-wicket before going for four.
But next ball lively seamer Marsh got a delivery to leap off a length and it took Bell's glove before lobbing gently to Clarke, who will retire from international duty after this match.
Cook, however,went on to complete a 119-ball fifty, including nine fours, after hitting three well-struck boundaries in one over from Marsh.
