Australia’s Mitchell Starc sparked an England collapse at Lord's on Sunday before Ben Duckett was given a controversial reprieve as the hosts maintained their bid for a remarkable win in the second Ashes Test.
England were 114/4 at stumps on the fourth day, still needing a further 257 runs to reach a target of 371.
If achieved, that would be England's second-highest run-chase to win a Test behind their 378 against India at Edgbaston last year.
Left-handed opener Duckett, who made 98 in the first innings, was 50 not out and captain Ben Stokes unbeaten on 29, with England looking to level the five-match series at 1-1.
Starc, recalled following Australia's thrilling two-wicket win in last week's first Test at Edgbaston, reduced England to 13/2.
And they were in dire straits at 45/4 when Australia captain Pat Cummins struck twice in an over.
Duckett then appeared to fall for exactly 50 when he miscued Cameron Green to fine leg, where a sliding Starc held a fine catch only for the ball to touch the turf.
Third umpire Marais Erasmus, to Australia's visible despair, disallowed the dismissal because Starc had grounded the before he was in control of his body and so had not completed a fair catch.
"I have no idea on the rules, we've seen different ones in this game and different decisions have been given," England batting coach Marcus Trescothick told the BBC.
'Finely balanced'
At the close, England could still think of a win that would be almost as improbable as their one-wicket victory over Australia at Headingley in 2019, when Stokes hit a stunning unbeaten hundred as they chased down a target of 359.
"It's finely balanced," added Trescothick. "We didn't want to lose the wickets at the top order but we faced some beautiful balls, it's not out of the question yet."
Starc had Zak Crawley 'strangled' down the legside, wicketkeeper Alex Carey holding a fine diving catch.
But there was no denying the left-arm fast bowler's skill when he cleaned up Ollie Pope with a 90 mph inswinger that started outside off and smashed into middle stump.
And Starc would have had Duckett lbw for five but for a review that fell narrowly in the opener's favour.
But Cummins had Joe Root fending to David Warner at first slip for 18 as the England star became the latest batsman this match to fall to the short ball.
Three balls later Cummins, looking to become the first Australia captain in 22 years to win an Ashes series in England, bowled Harry Brook with a superb delivery that straightened off the seam to hit the top of off stump.
World Test champions Australia ripped through England's top-order despite being without Nathan Lyon.
The off-spinner, appearing in his 100th consecutive Test, was not on the field having surprisingly limped out to bat at No 11 despite suffering a severe calf injury while fielding on Thursday.
Yet for all the 35-year-old's bravery, Lyon was struggling to run. And with runners no longer allowed in Tests, Starc repeatedly refused singles to a deep-set field in order to protect Lyon.
The courageous Lyon's defiant four off 13 balls ended when he holed out off Stuart Broad, the veteran paceman taking 4/65 as Australia were dismissed for 279 in their second innings.
Lyon departed to a pat on the back from Stokes, who bowled 12 overs straight through, and a standing ovation from the crowd. His gutsy effort was also a testament to Australia's regard for England's run-chasing ability.
In the past year, Stokes's men have achieved three of England's highest fourth-innings run chases.
The most any side have made to win in the fourth innings of a Test at Lord's, however, is West Indies’ 344/1 in 1984.
Australia resumed on Saturday morning on 130/2 but then lost three wickets for 10 runs in slumping to 197/5 as a trio of batsmen were bounced out.
Usman Khawaja, who made 141 and 65 at Edgbaston, started the slump when out for 77 after spooning a Broad bouncer straight to fine leg.
Steve Smith, fresh from a hundred in Australia's first-innings 416, then hooked a Josh Tongue bouncer straight to deep backward square.
AFP