Broad, whose magic spells are fast becoming one of the grandest sights in British sport, finished with his best overseas figures of six for 17 to roll the shell-shocked Proteas for 83 on day three in Johannesburg.
At one stage the 29-year-old had taken five wickets in 31 balls for just one run – and that off a dropped catch.
England required just 74 to take the series 2-0 with one to play, with captain Alastair Cook making 43 of them before his departure allowed Joe Root to score the winning runs.
Cook’s side may arrived here in sixth place but with a young, hungry and improving side they can have realistic designs on the summit themselves.
An early finish hardly seemed on the cards when the sides reconvened at the Wanderers after two enthralling, nip-and-tuck days, but Broad is a man capable of making things happen in a hurry.
His second success of the day saw him overtake Bob Willis’ 325 to become the country’s third highest wicket-taker, but as a purveyor of decisive moments England may never have had better.
South Africa started their second innings just 10 behind, but Broad’s afternoon assault shifted the ground beneath their feet.
The left-handed openers proved easy prey as he probed off stump from around the wicket.
Dean Elgar followed one as it nipped off the pitch and Stiaan van Zyl, having already been dropped at slip, popped a catch to gully after Broad persuaded one to straighten.
His third wicket was the big one, captain and star batsman AB de Villiers dismissed for a duck as a Broad in-cutter took a thick inside edge on its way into Jonny Bairstow’s gloves.
Despite catchers lining up in the cordon and gaps appearing all over the field, Broad proved impossible to score from.
The double centurion of Cape Town, Hashim Amla, was the next to fall at his hand, turning one off his pads and seeing James Taylor take a fine reaction catch low down at short-leg.
Taylor deserves a hearty slap on the back for his smart reactions, but he would have to accept being an extra in Broad’s show.
Temba Bavuma suffered a worse fate, weaving away from a short ball that failed to get up and cannoned into the stumps via a glove.
When Broad was finally persuaded to take a breather after 10 overs, the Proteas’ pain did not stop.
Taylor produced a second stunning catch under the helmet as Steven Finn dismissed Dane Vilas and Ben Stokes proved too good for Chris Morris and Kagiso Rabada.
Morris was castled by a gorgeous inswinger and Rabada, who had earlier taken a maiden five-wicket haul, handed Bairstow his ninth catch of the Test.
James Anderson belatedly joined the fun in his first over after the break, Hardus Viljoen trapped plumb lbw, but it was fitting for Broad to wrap things up.
He surprised Faf du Plessis with a rising delivery and flung himself forward to grab a diving return catch.
Elgar spared the hosts the ignominy of a 10-wicket defeat, prising out Alex Hales (18) and Nick Compton (nought), before Cook edged Morris with three required.
Root whipped Elgar for four to kick off the celebrations, having started the day in disappointment.
He had started the day 106 not out but added just four before becoming the first of 18 wickets.
That left Bairstow to make a crucial 45, dragging England to 323 and a slender lead.
He was last man out but his dogged knock left a platform that Broad sprung from in scintillating fashion.
Root’s unbeaten 106, his ninth century and second overseas, was a gem of an innings in the kind of circumstances that distinguish players or true class from mere pretenders.
Despite being in clear pain at times, he faced down a snarling Proteas pace quartet – all of whom topped 90mph on day two of the third Test – and counter-punched brilliantly on an awkward Wanderers track that never allowed the batsmen to settle.
Root’s knock, aided by a dashing half-century from Ben Stokes, saw England reach 238 for five before bad light stopped play in reply to 313 all out.
It had threatened to be much worse.
Root arrived with the scoreboard at 22 for two and was still holding up his end at a worrying 91 for four.
But the Yorkshireman helped turn the tide with a fifth-wicket stand of 111 in 95 balls alongside Stokes, and carried on to reach three figures for the first time in 13 innings.
Thanks to his efforts, which seemed hampered either by long-standing back problems or some new strain, England are harbouring genuine hopes of a vital first-innings lead.
The day began with South Africa adding 46 for their final three wickets.
That was more than England had bargained for, as was the unexpected sight of James Anderson being removed from the attack after a third warning for running on the pitch.
Stokes took the final wicket immediately after being asked to complete a disgruntled Anderson’s over, but England’s trials were just beginning.
Struggling opener Alex Hales fell for one, pushing Kagiso Rabada to the waiting AB de Villiers, and Alastair Cook (18) handed debutant Hardus Viljoen a prize scalp from his first Test delivery.
England might easily have lost a third before the break, Root and Nick Compton penned back and beaten a handful of times by the pace and aggression of Rabada and Viljoen.
Compton’s 50 stand with Root was an important one but his was a curious knock in two distinct parts.
He eked out six runs in 46 balls before being dropped by AB de Villiers, then added 20 in 21 before prodding Rabada to second slip.
Root was finding ways to stay in a tough contest, working deftly into the leg-side and punishing width, but James Taylor’s stay was brief and jittery.
He fell to the first ball of a new Morne Morkel spell, Temba Bavuma taking a tough short-leg chance by parrying the initial attempt.
The crowd sensed a key moment as Stokes, England’s double centurion in Cape Town, entered the Bullring and Morkel responded with a pair of brutish deliveries.
The first rattled the splice and the second reared angrily, thudding the glove in front of Stokes’ face.
The resulting counter-attack was utterly compelling.
In the next eight overs before tea, Stokes and Root piled on 70 runs.
Stokes’ second scoring shot was a powerful pulled six off Rabada, his second a chancy edge for four.
Root was less frenetic but equally effective, using South Africa’s added pace to deflect and steer his shots to the ropes.
There were moments of good fortune as well as high class, Stokes top-edging a pull high over the wicketkeeper’s head and Root flirting with the gully fielder as he reached the first half-century of the match.
Runs continued to flow after the restart, Stokes staying in fifth gear and Root happily swinging at anything short despite his growing discomfort.
Stokes’ half-century came in just 46 balls, with a particularly powerful blow off the bull-like Viljoen.
South Africa must have feared a repeat performance from Stokes after his Newlands spectacular, but he was cut off in his prime.
Morkel cramped him for room with a back-of-length ball on leg stump and gathered a loopy return catch off the top edge.
Root, whose closest calls had come from tight singles, forged on without his accomplice and reached his hundred with a delightful cover driven four, one of 17.
He unleashed a telling 100-watt smile and was congratulated by partner Jonny Bairstow, who had earlier spurned the chance to equal the Test record of seven catches in an innings by grassing Morne Morkel.
Set a decent target after Zimbabwe were restricted to 163-7, Bangladesh comfortably reached 166-6 with eight balls in hand to take 1-0 lead in the four-match series.
Zimbabwe made regular breakthroughs but could not stop the run flow as Tamim Iqbal (29), Mushfiqur Rahim (26) and Shakib Al Hasan (20 not out) all made decent contributions to Bangladesh’s victory.
Sabbir faced 36 balls and hit four boundaries and a six, taking advantage of a dropped catch by Vusi Sibanda off Sean Williams when he was on 36.
Earlier, Hamilton Masakadza and Sibanda put on a record 101-run opening partnership for Zimbabwe after skipper Elton Chigumbura won the toss and opted to bat first at the Sheikh Abu Naser Stadium in Khulna.
Masakadza hit 79 off 53 balls, equalling his own national record for the highest individual innings, while Sibanda struck 46 off 39.
Bangladesh did well to restrict Zimbabwe in the final few overs, despite Masakadza and Sibanda breaking the team record, set by Sikandar Raza and Chamu Chibhabha’s 100-run partnership against Afghanistan in October.
Bangladesh had to wait until the 12th over for their first breakthrough when Sibanda was finally caught at square leg by Tamim Iqbal off Shakib Al Hasan’s delivery.
Shakib dropped Masakadza on 73 off Mustafizur Rahman, but debutant wicketkeeper Nurul Hasan ensured it did not cost the team much by ending his innings six runs later with a brilliant throw for a run out.
Shortly thereafter, Rahman bowled Chigumbura and Luke Jongwe off consecutive balls to finish with 2-18, before Al-Amin Hossain grabbed two wickets in the final over.
Zimbabwe had to settle for a relatively modest score, picking up just 21 runs for five wickets in the last four overs.
The second match of the series will be held at the same ground on January 17.