At the close of a remarkable second day, Australia were 168 for seven in their second innings -- a lead of just 23 runs -- with three wickets standing and three days left to play.
Middlesex fast bowler Finn, recalled for his first Test since 2013, had taken five wickets for 45 runs in 13 overs following his two for 38 in Australia's meagre first innings 136.
Opener David Warner struck a defiant 77 today, one of only three double-figure scores so far in Australia's second innings.
A mere 19 matches in the 138-year history of Test cricket had ended inside two days, with the last occurrence in England when the West Indies were beaten by an innings and 39 runs at Headingley in 2000.
More From This Section
But a side strain suffered midway through his ninth over by James Anderson -- on his 33rd birthday -- hampered England's hopes of adding to that list.
The first two days of this match have witnessed a remarkable turnaround from England's humiliating 405-run defeat in the second Test at Lord's which saw Australia level the five-match series at 1-1.
But whereas that clash had taken place on a largely placid pitch, the course of this Test owed much to Australia captain Michael Clarke's decision to bat first on a seaming surface offering sideways movement.
Anderson, England's all-time most successful Test bowler but wicketless at Lord's, led the way with an Ashes-best six for 47 in a first innings that lasted a mere 36.4 overs.
Veteran opener Chris Rogers, whose 52 had been the lone batting highlight of Australia's first innings, fell for six when he was lbw to paceman Broad.
Finn was infamously deemed "unselectable" by then England one-day coach Ashley Giles when sent home early from the woeful 2013/14 tour of Australia which included a 5-0 Ashes defeat.
But he was near unplayable at times today as he dismissed both Steven Smith and struggling Clarke for the second time in the match.