The tourists must therefore make their highest total of a sorry series if they are to escape an Ashes whitewash in this final Test.
Rogers (114no) took the respective century count this winter to an astonishing 10-1 in Australia's favour when he cut part-time off-spinner Kevin Pietersen for his 14th four.
George Bailey fell four short of a second Test half-century, in a stand of 109 with Rogers which helped the hosts to a lunchtime 248 for seven and an overall advantage of 419 on day three here.
Bailey's departure, caught at deep square-leg off Stuart Broad, merely gave Brad Haddin the opportunity to try to pass 50 for the seventh time in the series.
He soon made it clear he was in a hurry to do so as well, with a declaration presumably looming.
Haddin's eyes lit up a little too much when he spotted a Scott Borthwick full toss, though, and he missed the slog-sweep to be bowled off-stump and fall seven runs short of a 500 series aggregate.
Mitchell Johnson also went, for just four, when Ben Stokes clean-bowled him just before lunch - the all-rounder's eighth wicket of the match.