Broad came into this fourth Test with just five wickets in the series, amid calls from some to leave him out as England sought to avoid a 5-0 whitewash, while all-time national record run-scorer Cook had failed to muster a half-century in his six innings Down Under.
Mixed feelings therefore inevitably prevailed as, with the urn already gone, Broad burst into life first with 4-51 as Australia lost their last seven wickets for 67 to be bowled out for 327, and then Cook (104no) steered England to 192-2 alongside Joe Root (49no) on day two.
The opener was posting his 32nd Test hundred, having failed to pass even 50 in any of his last 11 attempts - dating back to his 243 against West Indies in England’s inaugural pink-ball match at Edgbaston more than four months ago.
Tom Curran had got England up and running in the morning session when, after a third-umpire no-ball call had chalked off David Warner as his maiden Test wicket the previous day, he bagged an even bigger fish in Steve Smith (76).
The prolific Australia captain had added only 11 to his overnight score, and his stand with Shaun Marsh (61) moved to exactly 100, when he fell to the first of three drag-ons as the hosts found trouble on this slow pitch.
Smith was trying to carve off-side runs but managed only to chop the ball back down on to his stumps, and it was not long before Mitch Marsh fell in almost identical circumstances to Chris Woakes.
Broad, so close to seeing off the elder Marsh for a first-ball duck on day one, finally did so - again in an uncanny near repeat when he overturned an initial not-out verdict for lbw.
That was the first of five wickets which would fall for 13 runs as Australia, without the injured Mitchell Starc here to bolster their tail, proved vulnerable after all.
Old stagers Broad and James Anderson (3-61) took advantage, Tim Paine edging an attempted pull down on to his stumps off the latter and Josh Hazlewood last man standing as three other tailenders amounted to very little.
England had given themselves similar opportunities at various stages of the first three Tests, but never managed to capitalise.
This time, Cook ensured the foothold was secured.
He lost his opening partner Mark Stoneman to a leading edge and one-handed return catch by Nathan Lyon, and James Vince did not appear to feel a thin inside-edge which was picked up by Hotspot and would have overturned his lbw dismissal to Hazlewood.
Cook rarely looked likely to pass up his chance once set - although he did have a moment of fortune when Smith dropped him at slip on 66 off Mitch Marsh.
Becalmed, he nonetheless moved above former Australia captain Allan Border up to ninth in the all-time global list of Test runscorers - and then, just before the close, pulled part-time leg-spinner Smith for his 15th four from 164 balls to reach three figures.
By then, he had acquired the assistance of Root in an unbroken century stand which put England in position to push for an elusive first-innings lead if they can only sustain this much-improved performance.