Asked to face a tricky two-over period before lunch on day four, England lost Ben Duckett for four when he edged a Cummins delivery to second slip.
The opener's dismissal summed up a horror tour for the left-hander, who is yet to pass 30 and has quickly been exposed over his refusal to leave the ball.
Duckett's exit came after Australia were bowled out for 349 in their second innings, falling relatively quickly after starting day four at 4-271.
Saturday morning was one of missed opportunities for Australia's batters, with Travis Head (170) falling short of a double century and Alex Carey (72) missing out on successive hundreds.
Josh Inglis was also caught in the slips for 10, raising further questions over his spot in Australia's side with Steve Smith expected to return in Melbourne.
But still the hosts have ensured a retention of the Ashes is in sight, with the West Indies' 418 against Australia in 2003 the highest ever fourth-innings chase.
England's current best effort is 373 against India earlier this year, albeit achieved with five wickets in hand.
If Australia were to win on Saturday and take a 3-0 series lead, it would mark the fastest an Ashes series has been decided since 1921 with just 10 days played.
With the potential of an historic day in Adelaide, Head begun the morning on 142 and Carey 52.
Again, England were guilty of making it easy for Head, feeding him outside off stump and playing with one slip, no gully and a deep backward point.
He brought up 150 when he cut Brydon Carse to the third-man boundary for four, before two thunderous blows off the quick through mid wicket and cover-point.
But with a maiden Test double century in sight, Head was out when he pulled Josh Tongue (4-70) to Zak Crawley at deep backward square.
Carey also looked on track for his second hundred of the match, where he could have become the first Australian wicketkeeper to record two tons in a Test.
After beginning his day with a cut shot for four off Carse (3-80), his wicket came out of nowhere when he nudged Ben Stokes to leg slip.
Inglis came and went for 10 when caught at slip, no doubt prompting consternation over whether he is squeezed out for Usman Khawaja to stay on Boxing Day.
Notable too was the fact Stokes bowled the first over of the day for England, after coaching staff said he was "too knackered" to bowl with the series on the line on Friday.
Stokes had offered one of England's few resistances of the series with his 198-ball 83 in the heat in the first innings, but the tourists could desperately have done with their captain bowling.
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