Steve Smith's patient 71 was the backbone of Australia's 295 in reply to England's 283 and a ninth-wicket partnership of 49 between Pat Cummins and Todd Murphy edged the tourists ahead.
Australia, 2-1 up in the series and bidding to win the Ashes in England for the first time since 2001 were bowled out just before the close of play.
"It's ebbed and flowed the whole way, we're pretty disappointed, the wicket itself was good, played pretty nice," Smith told the BBC.
"A few of us got good starts but couldn't go and get a big score. Twelve runs in front, it's a one innings game from here pretty much."
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Resuming in the morning on 61-1, Usman Khawaja and Marnus Labuschagne had adopted a no-risk approach, adding only 21 runs in the first 12 overs before the latter was brilliantly caught by Joe Root at first slip off Mark Wood for nine.
Labuschagne edged Wood between wicketkeeper Jonny Bairstow and Root who dived to his left and plucked the ball out of the air one-handed.
England captain Ben Stokes did not have the option of using his main spinner Moeen Ali, who sustained a groin injury while batting, but his quick bowlers produced disciplined spells to keep the hosts in the game as Australia reached 115-2 at lunch.
ENGLAND'S AFTERNOON
Stuart Broad struck twice after the interval and England dominated the afternoon session.
Broad trapped Khawaja lbw for 47 and Travis Head, on four, edged the fast bowler to Bairstow as England sensed an opportunity to seize the initiative.
James Anderson claimed his first wicket when Mitchell Marsh, on 16, dragged the ball onto his stumps and Alex Carey lifted Root for six before perishing next ball when he miscued a drive into the hands of Stokes at short extra cover.
Mitchell Starc fell to a short ball from Wood for seven, Ben Duckett taking a simple catch at long leg, and Australia were reeling at 186-7 at tea.
Smith, who narrowly avoided being run out for 42 when the third umpire decided Bairstow had not gathered the ball cleanly before breaking the stumps, reached another test fifty before trying to sweep Chris Woakes, the ball flying high off the edge of his bat and Bairstow running back to take a good catch.
Australia's tail wagged hard, however, and Murphy hit Mark Wood for three sixes, his first in top-level cricket before Woakes returned to dismiss him lbw for 34.
Cummins was the last man out for 36, superbly caught by Stokes on the boundary to end another dramatic day in an enthralling series.
"It's a very level game," Anderson said. "Getting into the position we did, we're a little disappointed they got past us.
"Pat and Murphy batted really well in the end. All in all, a really good day for the bowlers. It felt like a slow pitch and we had to slam it to get anything out of it."
