Set an improbable 504 to win, England were 251 for five at stumps on day four, still 252 runs behind with just five wickets in hand on a heavily cracked Perth pitch.
Stokes, playing just his second Test, was providing resistance on 72, with wicketkeeper Matt Prior on seven and all of England's top order back in the pavilion.
Australia will be heavily favoured to mop up England's tail on the final day and regain the famous Ashes urn for the first time since 2007 after their big wins in Brisbane and Adelaide.
After a catastrophic day for the tourists on Sunday, when they conceded a big lead to the Australians, things only got worse for them on day four.
Centurion Shane Watson and George Bailey humiliated the England attack before Australia declared on 369 for six, making them the first team to be set a target of more than 500 in three successive Tests.
England's chase then got off to a disastrous start when captain Alastair Cook fell for a golden duck, the first of his 100-Test career, from the first ball of their second innings.
Fellow opener Michael Carberry again looked solid, but was trapped lbw by Shane Watson for 31, while Joe Root defended grimly for 125 minutes before being caught behind off Mitchell Johnson for 19.
Root called for a review, but it showed a clear edge through to wicketkeeper Brad Haddin, who took a fine diving catch.
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