Spinner Nathan Lyon and paceman Mitchell Johnson spearheaded a rousing Australian fightback after trailing by 51 runs on the first innings to skittle the tourists for 179 in 61 overs.
Lyon finished with five for 50 and claimed his 100th Test wicket, while man-of-the-series Johnson took three for 25 and now has 31 wickets in the series.
The Australians, needing 231 to win, reached stumps on 30 without losing a wicket with Chris Rogers on 18 and David Warner on 12.
Although the highest successful run chase at the MCG is England's 332 for seven in 1928, since the current 'drop-in' pitch was first used at the ground in 1996 the highest successful fourth innings run chase since then is 183 by South Africa in 2008.
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The Australians roared back into contention with a sensational bowling and fielding performance with Lyon and Johnson the heroes on a compelling day's cricket after they were all out for 204.
Johnson played a role in three of four dismissals to have the tourists tumbling to 87 for four in the middle session.
Cook, who celebrated his 29th birthday on Christmas Day, became the youngest player to score 8,000 Test runs before he was trapped leg before wicket by Johnson.
Carberry went the same way to Peter Siddle for 12 off 81 balls before Joe Root was run out by a brilliant piece of fielding from Johnson.