India pulled off a dramatic record run-chase to knock the reigning champions, Australia, out of the World Cup, led by an outstanding century from Jemimah Rodrigues.

Rodrigues was dropped three times by Australia, on 52, 83 and 106, eventually finishing unbeaten on 127 as the seven-time world champions uncharacteristically cracked under the pressure of defending 338. Running back from mid-off, Tahlia McGrath spilled the third of those chances from Rodrigues in the 44th over, while Annabel Sutherland sent down two wides in a wayward 48th over which cost 15 runs.

It was left to Amanjot Kaur to reap the rewards of Rodrigues’ staying power, striking the winning runs from Sophie Molineux’s 49th over with nine balls to spare.

The historic run-chase – the highest ever achieved in women’s ODI cricket – had hung in the balance after Harmanpreet Kaur miscued a pull to midwicket in the 36th over, out for a run-a-ball 89 with 113 runs still needed from 88 balls.

Deepti Sharma was then run out after hesitating over a quick single, while Richa Ghosh smashed two sixes but sent up a catch to short third with the equation down to 29 off four overs.

Rodrigues, though, remained unbeaten to the end, as India waltzed into a home World Cup final with a five-wicket win, delighting the Mumbai crowd. They face South Africa at the same ground on Sunday.

“It was really hard these last 12 months, but this feels like a dream and it’s not sunk in yet,” Rodrigues told Sky Sports – a reference to a tricky period in which she has been in and out of the India XI. Less than two weeks ago, she was dropped for the group-stage match against England, while in the semi-final she was told only five minutes before being sent out that she would be batting at No 3.

“I cried every day throughout this tour,” she said. “I was not doing well mentally, going through a lot of anxiety, then being dropped was a big challenge for me.

“ But [this innings] was not for me to prove a point, it was for me to win this match for India. We’ve always lost in crunch situations, and I wanted to be there to the end to take us through.”

India were clear underdogs going into this game, having limped through the group stages with losses to these same opponents as well as to South Africa and England. That status was only underlined when opener Phoebe Litchfield hit a 77-ball century – the quickest ever in a World Cup knockout match – and India lost both of their openers in the powerplay, in dismissals which epitomised the two teams’ differing success with use of DRS this World Cup.

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Jemimah Rodrigues hits out on her way to a brilliant unbeaten 127 against Australia. Photograph: Francis Mascarenhas/Reuters

Shafali Verma, catapaulted into the XI after being approved as an injury replacement three days ago, lasted five balls before being rapped plumb in front by Kim Garth – but opted to burn through a review anyway.

Australia then successfully overturned a not-out caught-behind decision against Smriti Mandhana, after replays showed the tiniest of spikes on UltraEdge. Mandhana shook her head in disbelief before walking off.“We created pressure, we created opportunities, we just weren’t able to capitalise,” Alyssa Healy, who also dropped Rodrigues, told Sky Sports. “That’s something that Australia really prides themselves on, and we let ourselves down in that regard.” Earlier, Australia’s total of 338 was built around that maiden World Cup hundred from the 22-year-old Litchfield, a quiet half-century from Ellyse Perry and a much noisier one from Ash Gardner.

Litchfield unfurled her trademark reverse-sweep, although there was a nervous moment when she tried it on 62 and was given out on-field after the ball ended up in the hands of backward square leg. Litchfield, though, sent the decision upstairs and was reprieved when replays showed it had been a bump ball.

That allowed her to progress to her ton eight overs later – but she was bowled trying to ramp Amanjot Kaur in the 28th over, and India fought back in the middle overs to take four Australian wickets for 45 runs. Perry, who had looked set to join Litchfield in the centurion club after scoring her first half-century of the tournament, joined in with the collapse, bowled by a Radha Yadav ball which skidded on to her stumps.

Gardner muscled four sixes and brought up a 41-ball fifty to ensure Australia added 38 runs from the final four overs – but she was run out after a muddle with Kim Garth in the 49th, while Deepti Sharma took two wickets with successive deliveries in the 50th over as Australia were bowled out off the penultimate ball of the innings, a limp finish which eventually came back to haunt them.