Australia's Glenn Maxwell celebrates after their win in the World Cup match against Afghanistan in Mumbai, India, on Tuesday. AP

Glenn Maxwell's stunning 201 not out guided Australia into the semi-finals of the World Cup as the five-time champions snatched a sensational three-wicket victory from the jaws of defeat against Afghanistan in Mumbai on Tuesday.

At 91-7, chasing 292, Australia were all set to join title-holders England, 1996 champions Pakistan and 1992 winners Sri Lanka as yet another notable Afghanistan scalp at this World Cup.

Crucially, however, recalled all-rounder Maxwell was dropped soon afterwards on 33 when Mujeeb Ur Rahman somehow floored a simple chance at short fine leg off fellow spinner Noor Ahmad.

"It would've been nice if it was chanceless, I've lived a very charmed life out there," said Maxwell, appropriately nicknamed 'The Big Show'.

He made Afghanistan pay with an astounding 128-ball innings featuring 21 fours and 10 sixes as he became just the third batsman after New Zealand's Martin Guptill and West Indies' Chris Gayle to score a World Cup double century.

Maxwell ended the match in style with nearly three overs to spare when he slammed Mujeeb for another six.

The 35-year-old Maxwell's highest one-day international score was just his fourth century in 136 matches at this level, but second of the tournament after he scored the fastest hundred in men's World Cup history, off just 40 balls, against the Netherlands.

Australia captain Pat Cummins was 12 not out but played his part by holding firm in an unbroken eighth-wicket stand of 202.

There was one more twist late on in a dramatic encounter when Maxwell, on 146, suffered a violent attack of cramp as he hobbled through for a single.

But following on-field treatment a still-struggling Maxwell continued to pulverise Afghanistan's attack with a dazzling array of shots.

These included an extraordinary reverse hit for six off paceman Azmatullah Omarzai -- earlier denied a hat-trick by Maxwell.

Afghanistan, despite this defeat, remain in last-four contention but they will need to beat second-placed South Africa and hope other results go their way to qualify for the knockout phase.

It had seemed the night would belong to Ibrahim Zadran, whose 129 not out in a total of 291-5 was the first hundred by an Afghanistan batsman at a World Cup.

Thar left Australia needing to better their previous highest winning score to win a World Cup match of 287 against New Zealand in a 1996 quarter-final in Chennai

But opener Travis Head fell for a duck, caught behind off a superb Naveen-ul-Haq delivery that seamed away sharply, with the paceman also having Mitchell Marsh lbw for 24.

Australia's 43-2 then became 49-4 as Omarzai struck with successive deliveries, clean bowling veteran opener David Warner following the left-hander's ugly heave across the line before Josh Inglis edged to first slip.

Maxwell survived the hat-trick, nicking short of wicketkeeper Ikram Alikhil.

Australia were 69-5 when a mix-up with Maxwell saw Marnus Labuschagne run out by Rahmat Shah's direct hit from midwicket before leg-spinner Rashid Khan removed Marcus Stoinis and Mitchell Starc as the crowd roared Afghanistan on.

But then came Maxwell's big reprieve, with Australia 112-7.

Earlier, opener Zadran took Afghanistan to their highest World Cup total as they topped their 288 they made against the West Indies at Headingley in 2019.

Australia initially kept Afghanistan in check after losing the toss, albeit they struggled for wickets with Zadran completing his fifth hundred in 27 ODIs, off 131 balls, including seven fours.

Zadran, however, accelerated after going to three figures with Rashid making a dashing 35 not out during a quickfire unbroken stand of 58.

Agence France-Presse