South Africa left the World Cup with a defiant raspberry, defeating Australia by 10 runs in another Old Trafford last-over thriller to set up a semi-final between England and Australia on Thursday.

Tension had mounted after the result zipped across from Headingley, which meant Australia had to beat South Africa to finish top of the table – fail, and India would tuck into the freshly prepared New Zealand fattened calf, tender and soft, and send the reigning champions instead to Birmingham.

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In the royal box at Wimbledon, Eoin Morgan licked his lips and ordered another Pimm’s: meeting Australia at Edgbaston in a semi-final was a far tastier proposition than at Lord’s next Sunday. The very last match of the long round-robin stage suddenly mattered.

Australia had chased only twice in this World Cup, once against Afghanistan, and once, badly, against India. Patched together and limping, and after a lacklustre bowling performance, a target of 326 looked unlikely. At 119 for four at nearly halfway, it seemed off-piste.

But there was David Warner, brisk and businesslike, short little legs prickling with determination, bursting chest-proud during the national anthem, slapping the ball high and long – for midwicket sixes and back over the bowlers’ heads. Bang he went, passing 600 runs for the tournament; bang-bang: his third World Cup century; up he leapt, kissing the badge. Pluckily he took control of the chase, finding in Alex Carey a partner of some staying power, computing the field placings and picking the gaps.

Warner fell with 11 overs left and scoreboard pressure mounting, dancing down and darting at Dwaine Pretorius, before being brilliantly caught at mid-on by a soaring Chris Morris.

Carey, who reached his own 50 by ramping the ball over slip, took his chance, tearing into Imran Tahir. But South Africa held their nerve: Aiden Markram took a pressure catch at deep cover to dismiss Carey for 85 and Kagiso Rabada stole two wickets in the penultimate over, leaving Australia with just too much to do.

The omens were bad for Australia’s batting from the start. Warner just survived being run out in the first over, Aaron Finch flicked Rabada airily just short of third man and Warner played and missed. Then Finch fired a heavy backfoot biff to cover and 10 minutes later Usman Khawaja was limping off with a dodgy hamstring – when he returned at the denouement, he could barely run.

Steve Smith swivelled a four before being trapped lbw by Pretorius. Marcus Stoinis chunked a few boundaries without ever looking convincing, and soon he too was swallowing tablets from the physio’s magic shoulder bag following a recurrence of an old side strain.

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Then two moments of Quinton de Kock magic: a back-handed flick to hurry-on a throw from Rabada to run out Stoinis; and a springing left-handed leap as Maxwell swung at a bouncer.

Australia’s afternoon in the field was summed up by Mitchell Starc’s first over: he stroked his moustache, ran in for the first ball of the match and delivered a 90mph legside wide that flew over the boundary. Another wide followed two balls later, before Markram drove him twice between cover and point for four. The World Cup’s leading wicket-taker was out of sorts. Soon he’d be clutching at his knee uncomfortably as well. It went downhill from there.

Faf du Plessis made an immaculate 100 off 94 balls; Rassie van der Dussen, a jellyfish to the Australian spinners early on, found his mojo, out for 95 going for a six off the last ball of the innings. South Africa’s four highest scores of the World Cup have come in the last two games.

Starc snaffled two wickets at the end, equalling Glenn McGrath’s World Cup tally of 26. But there is no time to rest on any laurels; he and the rest of the banana walking wounded head to Edgbaston for what Aaron Finch called a “blockbuster” –one Australia could have done without.