Unbeaten India were due to face 2009 champions England at the Sydney Cricket Ground in the first semi-final, but heavy showers began early in the day and barely let up.
A minimum of 10 overs per side were needed for a result and the umpires called it off without a ball being bowled.
With no reserve day, the highest-ranked teams from the two groups moved into the final if play was not possible, putting India through to their first-ever decider.
"From day one, we knew we had to win all the games in case the semis didn't happen," said Indian captain Harmanpreet Kaur. "First T20 World Cup final means a lot for us. But as a team, we just want to play our best cricket in the final."
The rain eased for the second semi and defending champions Australia hit 134 for five off their 20 overs, with skipper Meg Lanning top-scoring on 49 and Nadine de Klerk taking 3-19.
But rain briefly returned at the interval, and the Proteas were set 98 from 13 overs to win under the Duckworth-Lewis-Stern method, but could only manage 92 for five. "It was extremely tense. Even in the last couple of balls it just wasn't safe. We felt we had a good total without it being great," said Lanning.
"India are a class side, they have been playing excellent cricket. But we have been improving each day," she added.
Big-hitting opener Lizelle Lee smacked a huge six to get the South African run chase underway but, tempted into another, holed out to Ashleigh Gardner on the boundary ropes.
Skipper Dane van Niekerk looked dangerous but pace spearhead Megan Schutt clean-bowled her for 12, then Lanning took a difficult catch to remove Mignon du Preez and leave South Africa struggling at 26 for three.
But Laura Wolvaardt (41 not out) and Sune Luus (21) got them back on track to set up an exciting finale with 32 runs needed off the last three overs, only to fall short.
Earlier, Beth Mooney and Alyssa Healy aggressively set about building a big total for Australia.
Healy fell after a quickfire 18 and fellow opener Mooney blasted 28 before being clean-bowled by de Klerk.
South Africa successfully put the brakes on the run rate and grabbed two more quick wickets before Rachael Haynes (17) and Lanning guided them to a winning total.
"It was very emotional to see the ladies like that. To come short every time is difficult," van Niekerk said of her devastated team.
A reserve day is allowed for the final and the lack of one for the semis was criticised by some players, with Kaur among those calling for change.
"Having reserve days in the future will be a great idea," she said.
England captain Heather Knight suggested before the match that "there will be a lot of pressure on the ICC to change" the rules.
"Really frustrating," she said after the washout. "Not how we wanted the World Cup to finish for us. No reserve day, no chance of getting play."
Cricket Australia chief Kevin Roberts said he asked the ICC about adding a reserve day with the Sydney weather looking ominous, but the request was denied. "We've asked the question and it's not part of the playing conditions and we respect that," he told Melbourne's SEN radio. "It gives you cause to reflect and think about how you might improve things in the future."