It was the hosts' first win of the tournament in front of a partisan crowd after their opening loss to South Africa in Group B.
After the start of the match was delayed by an hour due to rain, West Indies put Sri Lanka into bat and they made 50-3 - with captain Shashikala Siriwardena unbeaten on 18 - before another downpour halted play in the 11th over.
The hour-long delay meant that it was West Indies who came out to bat, set a revised target of 48 from eight overs.
Stars Stafanie Taylor (3) and Deandra Dottin (4) lost quick wickets and Sri Lanka then bowled two spinners, attacked by opener Tremayne Smartt (8) and Stacy Ann King (12) - both of whom also fell quickly.
They needed 12 off the final over, bowled by Eshani Kaushalya, but could only manage six - finishing on 42-8.
Chamani Seneviratna ended with figures of 2-4 while Inoka Ranaweera also had two wickets.
New Zealand bounced back from defeat to West Indies with a 22-run win over South Africa, also in Galle.
New Zealand compiled 151-5 in their 20 overs after South Africa won the toss.
Sophie Devine (59) and Frances Mackay (49) enjoyed a 94-run partnership for the third wicket to give them a platform after they had fallen to 24-2 in the fourth over.
South Africa openers Shandre Fritz (1) and Trisha Chetty (7) were out early then Susan Benade (24) and Alison Hodgkinson (31) failed to keep up with the run-rate before falling while looking to attack.
They were eventually bowled out for 129 with 11 balls remaining as Dane van Niekerk topscored for them with 34no.
With all four teams on two points, West Indies top the group from New Zealand and South Africa on net run rate.
The winners of the final fixtures will progress to the semi-finals.
In Group A England and Australia are on two points after opening wins over India and Pakistan.
The women's tournament is taking place concurrently with the men's World T20 in Sri Lanka.
Women's World T20 fixtures:
Group A: Australia v Pakistan (Galle) - Sep 29
Group A: England v India (Galle) - Sep 29
Group B: South Africa v West Indies (Galle) - Sep 30
Group B: Sri Lanka v New Zealand (Galle) - Sep 30
Group A: India v Pakistan (Galle) - Oct 1
Group A: Australia v England (Galle) - Oct 1
1st Semi-Final: Colombo - Oct 4
2nd Semi-Final: Colombo - Oct 5
Final: Colombo - Oct 7
South Africa beat Sri Lanka by 6 wickets
West Indies beat New Zealand by 7 wickets
England beat Pakistan by 43 runs
Sri Lanka beat West Indies by 5 runs (D/L)
New Zealand beat South Africa by 22 runs
- South Africa
- Sri Lanka
- West Indies
- New Zealand