Jos Buttler added just six runs to his overnight score before being bowled out on 70 by Australia's Pat Cummins, the top wicket-taker of the series so far.
Medium-pacer Marsh then completed his maiden five-wicket haul in test cricket by bowling out Jack Leach for 21, leaving Stuart Broad stranded at the other end without scoring.
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Australia are bidding to secure a 3-1 series victory after retaining the Ashes by winning the fourth test at Old Trafford, meaning England would at best only be able to draw the series 2-2. (Reporting by Hugh Lawson Editing by Toby Davis)
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