Skipper Alastair Cook (7), Ian Bell (16) and Kevin Pietersen (6) all fell in the middle session as the beleaguered tourists limped to tea on the third day, reaching 87 for three, chasing an improbable 448 runs for victory, with more than two days still to go.
Michael Carberry was unbeaten on 43 with debutant Gary Ballance not out seven at the interval but the demoralised tourists are virtually certain to become only the third England team in history to lose a series 5-0 to their oldest rivals.
Pietersen was athletically caught by George Bailey close to the bat off Harris to leave hapless England at 57-3 as their top order once again failed to fire.
Cook, who scored just seven in the first innings, played tentatively at a Johnson outswinger and was caught behind in the second over of the innings to give the left-arm paceman his 35th wicket of the series.
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The England skipper has scored just 246 runs at 24.60 in this series, in sharp contrast to the last tour of Australia in 2010/11, when he amassed 766 runs at 127.66.
Australia were dismissed for 276 in their second innings shortly after lunch with Chris Rogers top-scoring with 119.
Rogers, who scored 116 in the second innings of the fourth Test in Melbourne, was caught and bowled by debutant leg-spinner Scott Borthwick shortly after lunch. Borthwick finished with three for 33 off six overs.
Rogers earlier shared in a fifth-wicket partnership of 109 with Bailey, who top-edged Stuart Broad to Borthwick at deep square leg for 46.
Brad Haddin became the highest-scoring Australian wicketkeeper in a Test series with his breezy 28 off 40 balls.
The highest successful run chase at the SCG is Australia's 288 for two against South Africa eight years ago.