England, following on 332 runs behind, lost Adam Lyth (10), Ian Bell (13) and Joe Root (11) cheaply to put Australia in sight of a consolation win after losing the series.
Cook dug in with Jonny Bairstow (20 not out) to take England to 123 for three at tea, still 209 runs behind.
Lyth was caught by Michael Clarke at second slip off Peter Siddle, continuing a wretched run of form for the England opener who will surely be dropped after failing to pass fifty in the series.
Bell never looked comfortable before fending a snorting delivery from Mitchell Marsh to captain Clarke.
Root also struggled to cope with a fired-up Australia pace attack and he hooked a short ball from Mitchell Johnson to Mitchell Starc at long leg.
Cook took 27 balls to score his first run and batted with watchful intent.
He was quick to punish the rare bad balls, however, and hit nine fours on the way to his half-century which came after more than two and a half hours at the crease.
Earlier, Moeen Ali and Mark Wood extended their ninth-wicket partnership in England's first innings to 57 with a flurry of boundaries in the morning.
Wood struck six fours in his 24 before he attempted to pull Johnson and spooned a catch to Starc at mid-on.
Moeen, England's top-scorer with 30, edged Johnson's next delivery to wicketkeeper Peter Nevill to end the innings on 149 and Clarke, playing his final test, had no hesitation in enforcing the follow-on.