Cook, playing in an English-record 34th consecutive Ashes Test, and his 151st Test overall, batted with growing confidence after a string of low scores as the tourists reached 72 for one at tea.
The former captain had scored just 83 runs in his previous six innings of the lost series, but with the pressure off Cook got more into his stride.
At tea, England were trailing Australia by 255 runs with Cook unbeaten on 37 -- his joint-highest score of the series -- with James Vince not out 17.
The off-spinner is making a habit of sensational catches, particularly his flying effort across the pitch to dismiss Moeen Ali in Adelaide off his own bowling.
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England earlier ended Australia skipper Steve Smith's incredible three-year unbeaten run in Melbourne as they bowled out the hosts for 327.
England, already an irretrievable 3-0 down in the five-match series, claimed Australia's last seven wickets for 67 runs to wrap up the hosts' first innings shortly after lunch.
Smith had not been dismissed in a Melbourne Test match since Boxing Day 2014, and was on track for his third century of the series on a bat-friendly MCG pitch.
- Slice of luck -
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In the last three Boxing Day Tests, Smith scored 192 against India, 134 not out against the West Indies and an unbeaten 165 last year against Pakistan.
Smith threw his head back in annoyance as the bails flew into the air. With his dismissal, Smith has now scored 502 runs in the Ashes series at a stunning average of 125.50.
England got more return for bowling wide of the stumps when Mitchell Marsh also dragged on a Chris Woakes delivery for nine after his 181 in the third Test in Perth.
His elder brother, Shaun, lost a leg before wicket review on 61 off 148 balls to Stuart Broad.
It was Shaun Marsh's second half-century of the series after scoring an unbeaten 126 in the second Adelaide Test.
Pat Cummins was snapped up at slip off Broad for four on the second ball after lunch, before Lyon was lbw to Anderson for a duck to end the Australian innings.
Broad was the pick of the England bowlers with four wickets for 51 runs in his best bowling performance in more than a year.