At tea on the third day, England were 285 for seven in reply to Australia's first innings 566 for eight declared -- a deficit of 281 runs and still needing a further 83 to avoid the follow-on with just three wickets standing.
Moeen Ali was 38 not out and Stuart Broad five not out after Cook fell for 96.
Marsh, called up by Australia after fellow seam-bowling all-rounder Shane Watson was dropped following England's 169-run win in the first Test in Cardiff last week, justified his selection with two key wickets on Saturday.
Cook and Stokes put on 145 for the fifth wicket after the pair had come together following a top-order collapse on Friday that saw England slump to 30 for four in the face of hostile fast bowling led by left-armer Mitchell Johnson.
After Friday's play, which ended with England 85 for four, Johnson had tried to goad Stokes by questioning whether he would still continue to bat in the "aggressive" fashion that has been England's watchword under new Australian coach Trevor Bayliss.
Meanwhile Cook, 21 not out overnight, attacked when given the chance. He pulled a short ball from Josh Hazlewood for four and drove off-spinner Nathan Lyon straight for another boundary.
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