Scotland posted 5-160 after losing the toss in Hobart on Monday. Earlier, tweakers Mark Watt (3-12 from four overs) and Michael Leask (2-15 from four overs) proved their mettle for the team.
West Indies, who won the tournament in 2012 and 2016, could not find a partnership, reducing their chances of progressing to the next stage with defeat.
He was bowled out for 118 in 18.3 overs and only four players reached double digits. Veteran all-rounder Jason Holder scored 38 runs in 33 balls but was out with an assist.
This is the second major upset in as many days after Namibia beat Sri Lanka by 55 runs in Geelong on Sunday. Scotland took several difficult outfield catches, while Watt caught his three scalps while bowling from behind the crease.
West Indies lost 7-26 in the 13th over from the sixth over. Their evening was ended when Akil Hossein was run out in the 13th over despite Lesk requiring a second attempt to upset the stumps.
Earlier, Scottish opener George Munsey opened the innings with an unbeaten 66 off 53 balls. Scotland got off to a good start and posted 0-54 after the powerplay thanks to Munsey, who scored six of his nine boundaries in the first six overs.
He hit pacer Alzarri Joseph for three consecutive boundaries in the fourth over but struggled to regain his groove in the sixth over after a 45-minute rain delay.
Holder tightened the screws after rain at 2-14 but surprisingly bowled only three of his maximum four overs.
Group B teams, West Indies, Scotland, Ireland and Zimbabwe, are fighting to qualify for the Super 12 stage of the tournament.
They play each other once, with the top two teams advancing to the next stage.