An England win inside two days looked possible at one stage, but they will have to return today to finish the job.
Peter Nevill and Mitchell Starc shared a 64-run partnership before Finn again made the breakthrough, claiming Nevill for 59, with Australia still only 72 in front at 217-8.
Mitchell Johnson, and Australia, started the day brilliantly with an awesome over of bowling which saw off Jonny Bairstow (5) and Ben Stokes (0) thanks to some fearsome short-stuff.
He knows how important Anderson is to England, but had praise for Finn too.
England bowler Steve Finn celebrates a wicket 2015.
Critics of Clarke will point to his total of only 13 runs in this test, but Warner defended the captain as a “world-class” player.
“We’re still three wickets and some runs away from winning this game, so it is by no means done”, he said. But this wicket is proving hard for all batsmen with only Australia’s Chris Rogers and Root having passed 50 so far.
For England, the only real blot on an otherwise compelling second day came when Jimmy Anderson, in the middle of his ninth over, walked from the field, holding his left side below the rib cage.
England, after losing the toss, had bowled Australia out inside 37 overs on Wednesday, with James Anderson taking an Ashes-best six for 47.
His burst began with Adam Voges caught behind for 16.
Wicketkeeper Jos Buttler dropped a hard diving chance to dismiss Starc off Finn with 10 minutes left.
Moeen Ali did the bulk of the scoring and scored a fluent fifty under pressure.
It was a fitting way for spearhead Johnson to become the fifth Australian after Dennis Lillee, Shane Warne, Glenn McGrath and Brett Lee to take 300 Test wickets. Spinner Nathan Lyon has 3-14 and Johnson 2-39.
A win at Edgbaston would give England a 2-1 series lead with two more Tests to go and avenge the humiliating 405 run defeat in the second Test at Lords.
“As cricketers, you know the momentum can change quickly, so I don’t think we got ahead of ourselves”. But in the next over, Johnson put Australia right back into the match with two wickets for no runs in just three deliveries.