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However the demoralising defeat for Lyn Howell's side means they are nine points behind Ulster in fourth place.
The Dragons are still in touch with Ulster after they beat Munster 16-6 at Thomond Park.
Newly-appointed Wales coach Mike Ruddock paid tribute to his team's defensive play, they held out in the face of massive pressure in the closing minutes to to take the points.
With 48 points from 16 matches, they are just four points behind Ulster.
The Ospreys and the Edinburgh shared 66 points at Meadowbank in an entertaining match. Two late tries from the visitors gave them the points.