- René Joensen (84')
- Jonas Wind (85')
Denmark boss Kasper Hjulmand made nine changes from the team that beat Scotland 2-0 in midweek, giving some younger players a start in a game the Danes were expected to win comfortably but the Faroes held them scoreless through the first 45 minutes.
Wind had the ball in the net in first-half stoppage time but his effort was ruled out for offside after a VAR review.
The Faroes put on a tremendous defensive display before losing Rene Joensen to a second booking in the 84th, with Wind heading the winning goal a minute later.
The victory leaves Denmark top of the six-team group with a maximum 15 points, five ahead of second-placed Israel, while the Faroes are fifth on one point above Moldova on goal difference.