Needing quick wickets to have any chance of salvaging pride at the Melbourne Cricket Ground, England suffered a major blow when Cook put down two catches at first slip in the first half hour that would have removed both opening batsmen.
Though David Warner was dismissed cheaply for 25, Rogers combined with Shane Watson to guide Australia to 143-1 at the break, the hosts having added 113 runs in the session to punish a loose and resigned effort from England's bowlers.
Rogers was on 81 and Watson on 36, enjoying an 88-run partnership.
Having resumed needing 201 runs for victory, Rogers added only one run before nicking a Stuart Broad delivery behind and a lunging Cook put the chance down after wicketkeeper Jonny Bairstow failed to move for the catch that was rightfully his.
England heads dropped further when Cook put down the simplest of chances shortly after, when Warner drove recklessly at a Ben Stokes delivery to send the nick straight into the skipper's lap.
Mercifully for Cook, Warner added only three more runs, before slashing at another Stokes ball to be snaffled by Bairstow.
Rogers rode his luck to his half-century, moving to 49 when an inside edge whistled past the stumps and beat the keeper to run for four, but brought up the milestone with a lovely cut through the covers for two.
Watson combined with Rogers to punish the English attack, and after helping drive Australia past 100, the pair opened their shoulders in the last 40 minutes of the session.
Watson smacked Tim Bresnan for a pair of consecutive boundaries, while a comfortable Rogers carted the England bowlers for nine fours under bright sunshine.
Though victory and a 4-0 lead in the five-test series now appears a formality for Australia, no team has ever chased down more than South Africa's 183 in 2008 since drop-in wickets were introduced at the MCG in 1996.
Australia hold an unassailable 3-0 lead in the five-test series. The final match starts in Sydney on Jan. 3.