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Australia, seeking a fourth successive World T20 title, chose to bat and posted 148-5, with Elyse Villani making the first half century in a final.
But 18-year-old Windies opener Hayley Matthews hit a stunning 66 off 45 balls with three sixes, in a stand of 120.
The Windies won with three balls to spare and their men's team later completed a T20 double.
West Indies had lost all of their eight previous T20 internationals against Australia and in their first final they made a nervous start bowling at the iconic Eden Gardens.
Villani (52) seized on some wayward full tosses to reach her fifty from 34 balls.
Australia skipper Meg Lanning hit three successive fours in her 52 and the elegant Ellyse Perry dispatched two glorious straight drives for six in a cameo 28.
Having fought back by conceding only a single from the final over they bowled, the Windies made a slow start to their reply, scoring three from the first two overs.
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But Matthews and skipper Stafanie Taylor hit 16 from the fifth over, bowled by Perry, and recorded their century partnership in the 14th over.
Matthews was caught with 29 more runs required from 26 balls and Taylor's 59 ended with only five left to score from eight deliveries.
Three were needed from the final over and victory was secured after a routine run-out opportunity turned into an overthrow, to spark exuberant West Indian celebrations, joined by the men's team who were arriving for their final with England.
It continued a welcome resurgence for West Indian cricket, after the Under-19 team beat India to win the 50-over World Cup in February.
West Indies captain Stafanie Taylor:
"I've been waiting for this a long time and it has come at the right time.
"We didn't get the start we wanted but the batting has done it for us.
"It was fantastic to know that the men were with us. [Men's captain Darren] Sammy sent me a text this morning 'Staf you are going to do it, you girls you are going to do it'."
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Australia skipper Meg Lanning:
"Full credit to the West Indies, they came out with the bat, and we did not quite get the result we wanted.
"We felt we were a little short, 160 would have been nice, but if we bowled well we would have defended.
"It was a really exciting, tough road into the World Cup and I am proud of the effort that everyone has put in."
Former England batter Ebony Rainford-Brent on BBC Test Match Special:
"The Windies have brought power into this tournament.
"Hayley Matthews running down the wicket and hitting Ellyse Perry for six was probably the moment of the tournament.
"They have discipline in their game now. If this team can tighten up with the bowling and handle pressure they will be a dangerous force going forward."
West Indies claimed their first Women's World Twenty20 with a pulsating eight-wicket win over Australia in Kolkata.