Batting first, they reached 207 for nine, but held their nerve in the field to bowl out their opponents for 159.
Tremayne Smartt started the rot with three top-order wickets, while Stafanie Taylor and Deandra Dottin finished the job.
The result means that defending champions England's hopes of reaching the final remain out of their own hands.
Australia, unbeaten so far with four wins from four, will be in the final, but any of three sides could join them.
The West Indies will be finalists if they beat Australia in their final group match, but defeat will leave them open to being caught by England or New Zealand.
Those two sides go head-to-head on Wednesday in Mumbai, and the losers will be eliminated.
In the event of a West Indies defeat, however, it would come down to net run rate between the Caribbean outfit and the winners of England v New Zealand.
As it stands there is precious little between the sides: West Indies lead the way with an NRR of 1.13, England have 1.07 and NZ are on 0.95.
- Sports & Recreation
- Stafanie Taylor