The West Indies surpassed the challenging target with three deliveries to spare following a rollicking opening stand of 120 between skipper Stafanie Taylor and young Hayley Matthews.
The 18-year-old Matthews smashed 66 off 45 balls that included six boundaries and three sixes. Taylor scored 59 off 57 balls with six hits to the fence.
Both openers fell in the last five overs but Deandra Dottin, who had conceded just one run in the final over during Australia's innings, ensured victory with an unbeaten 18 off 12 balls.
It was the first win for the West Indies over Australia in nine T20 Internationals and the first time a team outside Australia, England and New Zealand won a major women’s world title.
Both Villani and Lanning scored 52 apiece for Australia, adding 77 for the second wicket after Alyse Healy had fallen in the second over for four.
Villani hit nine boundaries during her 37-ball knock while Lanning had eight fours including three off consecutive balls from Dottin.
Ellyse Perry boosted the total with a 23-ball 28 that included both sixes hit in the innings.
The West Indies made a brisk start as Matthews and Taylor hammered 45 runs in the six overs of PowerPlay before taking their team to 76 for no loss at the half-way stage of the innings.
Matthews reached her maiden T20I half-century in the 13th over with a six and four off Jess Jonassen, but appeared to be struggling with a leg injury.
Mathews finally fell in the 16th over when 29 were required off 26 balls as she pulled a ball from Kristen Beams to mid-wicket, giving the Australians their first break.
Taylor was dismissed in the 19th over, but not before finishing as the tournament’s leading scorer with 246 runs.
“We are certainly on top of the world,” said Taylor, who was later named the player of the tournament.
“I have been waiting for this for a long time and it has come at the right time. We did not get the start we wanted with the ball, but the batting did it for us.
“I had told the girls we needed to back ourselves against a top side like Australia and we did exactly that today.”
The closing stages of the match was watched by the West Indies men’s team which was due to play the final against England later in the evening.
“It was fantastic to see the men's team supporting us at the ground,” said Taylor. “In fact, Darren Sammy (the men’s captain) had sent me a text message this morning saying we were going to win. I am happy we proved him right.”
Lanning was understandably disappointed but conceded that the West Indies deserved to win.
“We did not get the result we wanted, but full credit to the West Indies for the way they came out with the bat,” the Australian captain said.
“I felt we were a few runs short because 160 would have been nice. But we could have defended 148 if we bowled well. That was not to be. But I am proud of the effort everyone put in to get us here.”
Matthews, adjudged the player of the match, was delighted. “Words can’t explain what I feel now,” she said, adding: “To be able to take the trophy home is a great feeling.”
West Indies beat Australia by eight wickets in Kolkata
Australia 148-5, 20 overs (Elyse Villani 52, Meg Lanning 52, Ellyse Perry 28; Deandra Dottin 2-33, Hayley Matthews 1-13, Anisa Mohammed 1-19)
West Indies 149-2, 19.3 overs (Hayley Matthews 66, Stafanie Taylor 59, Deandra Dottin 19 not out; Kristen Beams 1-27)