Australia whittled England's innings lead down to 51 before being dismissed for 204 in the morning session before England's openers Alastair Cook and Michael Carberry gave the tourists a secure start to their second innings.
In the process Cook, who celebrated his 29th birthday on Christmas Day, became the youngest player to score 8,000 Test runs.
Cook is now the sixth leading England run-getter behind Graham Gooch (8,900) and the 26th overall in Test cricket.
Earlier, Brad Haddin hit his fifth half-century of the series to be the last wicket to fall for 65 off 68 balls.
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Haddin put on 40 useful runs for the 10th wicket with Nathan Lyon, who remained 18 not out off 34 balls.
Haddin, one of the heroes of Australia's Ashes-series victory, was finally out when he top-edged an attempted pull shot off Jimmy Anderson and was taken by wicketkeeper Jonny Bairstow.
The number seven batsman has scored 390 runs and is averaging 65 and ranks second only to David Warner (466 at 77.66) as the leading scorer in the series.
Haddin and Lyon frustrated England for the opening 43 minutes of the third day and skipper Cook had his field well spread to stem the flow of runs as Australian reduced the tourists lead from 91 at the start of the third day.
Australia already have clinched the Ashes with big wins in the opening three Tests in Brisbane, Adelaide and Perth, but England have had a much better Test match in Melbourne and for the first time have an innings lead to put pressure on the home side.