Scotland needed 154 to win, a record chase, and were in it at 5-132 with two overs left.
Then medium-pacer Aaliyah Alleyne came to the West Indies' rescue. She took three wickets in the 19th over, including Darcey Carter on 59.
Scotland required 17 off the last over and were all out on the last ball for 146.
Only the West Indies and England are two from two in Group 2, with the top two making the semifinals.
Scotland had the West Indies on the ropes in both innings. Tight bowling and fielding reduced the Windies to 6-124 after 18 overs.
But another veteran, Stafanie Taylor, who missed the West Indies' opening win over defending champions New Zealand, rescued the batting effort.
Her unbeaten 47 off 19 balls with 34 runs in boundaries lifted her side to a competitive 6-153. They added 29 off the last two overs, 26 by Taylor.
Katherine Fraser starred in the field for Scotland. She took two wickets, including bowling Hayley Matthews, and a leaping catch on the boundary before effecting the run out of Shemaine Campbelle on 36.
Fraser and Carter then gave Scotland a brilliant flying start to the chase. But the Windies fought back with four wickets in 13 balls into the eighth over, and Scotland's run rate dropped below the required run rate for the first time.
Carter injured her leg in the fourth over and was dropped on 19. Struggling to run, she didn't hit a boundary after the fifth over until the 18th. Scotland went nearly five overs without a boundary.
But they stayed in the game thanks to poor bowling and fielding until Alleyne proved the difference.
On Friday, Ireland play New Zealand.
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