At the close of the third day, Australia were 108 without loss in their second innings, a lead of 362 runs.
England now needed to set a new ground record if they were to go 2-0 up in the series, with the highest fourth-innings total to win a Test at Lord's the West Indies' 344 for one in 1984.
Chris Rogers, who in the first innings of this match scored a Test-best 173, was 44 not out on his adopted home ground, having spent several seasons with Lord's-based county Middlesex.
He might have been out for nought when he edged James Anderson only for Adam Lyth, himself out for a duck in England's first innings, to make a mess of the chance at gully.
Earlier, England were dismissed for 312, 55 runs shy of avoiding the follow-on.
But Australia captain Michael Clarke, with two days left in the match, opted against making them bat again.
England's final total represented something of a major recovery after they had slumped to 30 for four on Friday in the face of hostile fast bowling led by Mitchell Johnson.
Both left-handed batsmen fell in similar fashion on Saturday, playing on to Mitchell Marsh.
The medium pacer took two for 23 after being recalled in place of Shane Watson following England's 169-run win in the first Test in Cardiff last week.
Left-armer Johnson, who took just the one wicket on Saturday, finished with three for 53 in 20.1 overs having looked like the bowler who took 37 wickets at under 14 apiece during Australia's 5-0 home Ashes rout of England in 2013/14.
But Stokes, whose 85-ball century against New Zealand in May was the quickest Test hundred ever seen at Lord's, showed good judgement after resuming on 38 not out.
He completed a 67-ball fifty when he off-drove Johnson for the eighth four of his innings, which also included a six.
But it was the lesser pace of Marsh which proved Stokes's undoing when he played on to leave England 175 for five.
The 30-year-old opener batted for nearly six hours and faced 233 balls including 13 fours.
Moeen Ali's 39 apart, the lower order offered little resistance with Johnson ending the innings when Stuart Broad (21) was caught by substitute Shaun Marsh, Mitchell Marsh's older brother, in the slips.
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