Australia hold a lead of 23 with three wickets remaining and three days to play after Finn, recalled after a two-year absence during which his shattered confidence and bowling action were restored, ran amok.
After capitulating to 136 all out on Wednesday and then facing a deficit of 145 after England made 281, Australia had no answer to Finn's pace, bounce and movement at a raucous Edgbaston.
He followed up his two wickets in the first innings with figures of 5-45 from 13 excellent overs as only a defiant 77 from David Warner and an unbeaten 37 from Peter Nevill held up England's victory bid inside two days as Australia closed on 168-7.
Australia's fragility was ruthlessly exposed by Finn after Chris Rogers (6) was trapped lbw to Stuart Broad.
Finn repeated his feat of the first innings by snaring Steve Smith and Michael Clarke in a superb spell either side of tea.
Smith perished for eight when he top-edged to wicketkeeper Jos Buttler and Finn had his tail up again immediately after the interval.
Australia captain's Clarke's poor run continued when he edged Finn to Adam Lyth at fourth slip and Edgbaston went wild again moments later when the paceman dug another short one into Adam Voges who fended straight to Ian Bell.
Mitchell Marsh saw off the hat-trick ball, allowing it to pass by off stump, but had made just six when his bails were sent flying by the towering Finn.
A pugnacious Warner still carried the fight, equalling the fastest half-century by either side in an Ashes test -- emulating the 35 balls compatriot Graham Yallop took in 1981 -- but the opener fell to James Anderson, who later went off with a suspected side strain.
Joe Root with 63 top-scored for England who were boosted by an entertaining eighth-wicket stand of 87 between Moeen Ali (59) and Broad (31).
Mitchell Johnson took his 300th and 301st test wickets in a rip-snorting second over of the morning, dismissing Jonny Bairstow and Ben Stokes.
(This story corrects the Marsh total to six in the ninth paragraph)
(Editing by Toby Davis)
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