Broad (six for 50) finished with 11 wickets in the match as the tourists, set 299 to win, collapsed from 147 for one to 224 all out.
David Warner appeared to be giving English supporters a proper reason to dislike him with his 71 runs as Australia threatened to reduce their Investec series arrears to 2-1 with one to play.
But from 168 for two, they lost their next five wickets for 13 runs to Broad and Tim Bresnan on the way to a terminal 3-0 deficit.
Warner, the pantomime villain of this summer after his errant punch at Joe Root in a Birmingham bar during the Champions Trophy, at last landed some apparently significant blows in the middle.
Yet his opening stand of 109 with Chris Rogers ultimately amounted to little as England completed an Ashes hat-trick for the first time since Ian Botham was turning matches on their head back in 1981.
On Monday, it was England's nearest thing to a latter-day version of Botham's all-round talents who drove the victory surge, Broad taking three wickets for two runs in 17 balls and all his six for 20 in 45.
It was fitting, after a hiatus as England battled bad light in the extra half-hour and were forced to take Broad's pace out of the attack, that he should return and have Peter Siddle caught at mid-off to be last out just after 7.40pm.
Bresnan was no slouch either, adding two wickets to a crucial innings of 45 which got England up to 330 all out this morning as Ryan Harris recorded career-best figures of seven for 117.
England were then unable to prevent the series' first three-figure opening partnership.
Rogers needed his fair share of fortune and, between the scores of three and 14, had four various scares ranging from lbw reviews to a dropped catch by Graeme Swann at second slip - an uncanny repeat of one of his first-innings escapes.
Warner completed his half-century from 74 balls when he timed Bresnan away off the back foot past cover for his eighth four to go with a six struck high over wide long-off from only the third ball bowled by Swann.
Rogers appeared set to follow his partner to 50 until he was caught at slip, one run short, when Swann got one delivery to grip.
Swann trapped Usman Khawaja too, lbw shortly after tea.
But it was when Bresnan struck in his first over - replacing the off-spinner after 15 in succession from the Finchale End and finding extra bounce to have Warner edging behind - that Australia's collapse took hold.
Broad produced an outstanding delivery to knock back Michael Clarke's off stump, and then had Steve Smith somehow defelecting a short ball down on to his stumps via a faulty pull.
Shane Watson and Brad Haddin both went lbw, to Bresnan and Broad respectively and each using up a review to decisions confirmed only by the 'umpire's call' margin of error.
After Broad then clean-bowled Nathan Lyon, for 25 minutes the unthinkable prospect loomed of a return tomorrow as the umpires repeatedly consulted their light meters - until a shaft of precious sunlight allowed England to go back, from spin at both ends, to Plan A.
Harris had taken two wickets in two balls this morning, but Bresnan's telling counter-attack helped the hosts add a combined 66 for the eighth and ninth wickets.
Harris saw off first Ian Bell and then the out-of-form Matt Prior for a golden duck.
Both were bowled, Bell (113) off an inside-edge as he tried in vain to jam down on a delivery which kept devilishly low and Prior unluckily off his elbow.
Bresnan responded, however, with three boundaries in succession off Jackson Bird as 60 runs, and three wickets, came in the first 10 overs of the second new ball.
Bell could add only eight on Monday, having faced 210 balls and hit 11 fours, to take his series tally to exactly 500 runs.
Harris bounced out Broad, caught off his left glove in the gully.
Nightwatchman Bresnan might easily have gone already, for 12, when he played no shot to Bird. But Aleem Dar decided not out lbw, and Hawk-Eye concluded umpire's call, the ball clipping the top of leg stump.
Bresnan was therefore still around as he, Broad and then Swann hurt Australia with important partnerships.
He eventually became Harris' seventh wicket, when he chipped back a caught-and-bowled. But there was more frustration still to come for Australia, when Smith dropped Swann at long-on off Lyon (three for 55), costing them another 13 runs they could ill afford before James Anderson was caught-behind off the off-spinner.
In the end, though, the margins did not come down to anything quite so narrow.