The all-imortant goal came from captain Rachel Hignett after 66 minutes in a match that saw both sides cancel each other out for long periods of the game.
Wales had lost their opening game 1-0 to hosts Moldova on Friday, whilst the Faroes had suffered a 10-0 drubbing to group favourites, France.
Wales will now play France on Wednesday, though the chances of qualification as one of the best second place sides looks slim.
France thrashed Moldova 8-1 in the afternoon's match, with braces for Léa Declercq, Kadidiatou Diani and Sandie Toletti.
The Wales team was:
1. Ffion Ashman
2. Rhian Cleverley - Yellow 60m
3. Katie Williams
4. Mali Summers
6. Hannah Lewis
8. Rachel Hignett (C) - Goal 66m
9. Emma Beynon
10. Allie Young
11. Ellie Leek
14. Rachael Nattrass
17. Samantha Quayle
12. Delyth Morgan
5. Hannah Williams
7. Sophia Rees
13. Bridie Davies (For 9 - 72m)
15. Georgia Evans (For 10 - 41m)
16. Stacey Webber (For 11 - 59m)
18. Sydney Hinchcliffe
Source - faw.org.uk
She Kicks - the online news service for women's football
