Opener Johnson Charles also played a big part in the result with a free-scoring 48 off 38 balls, which set the tone as they chased down Australia's total with 26 balls to spare after the visitors had compiled 265 for 7 off the 50 overs.
Carlos Brathwaite hit the winning run as the hosts, who were soundly beaten by Australia earlier in the series, recorded their second victory from three matches.
Australia have won two of four matches, while South Africa, the other team in the series, have one win from three games.
After nine games -- with each team playing six matches -- the top two qualify for the June 26 final.
West Indies' performance reaffirmed the competitiveness of the Caribbean team in the sport's shorter formats, a stark contrast to their recent struggles in five-day test cricket.
"It means a lot to us," captain Jason Holder said. "It just shows we can beat these teams. We just need to be consistent."
Australia, sent in to bat, promoted left-hander Usman Khawaja from number three to open in place of David Warner, who suffered a fractured finger on Saturday.
Khawaja made 98 before being run out, while captain Steve Smith chimed in with 74 as the pair put on a second-wicket partnership of 170 after Aaron Finch had been sent packing for a golden duck.
But it was slow going for Australia near the end of their innings, and Smith said his team's total was some 15 or 20 runs short of what he would have considered satisfactory.
"After 30 overs, we probably should have got a few more but credit to West Indies, I thought they bowled well," he said in an on-field interview.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Peter Rutherford; peter.rutherford@thomsonreuters.com)
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