Davies struck the opening goal at the EBB Stadium in the seventh minute when he beat Jamie Cumming with a looping header from Matthew Blake's free-kick.
Cullen doubled the visitors' lead three minutes from time when he fired into the top corner from the edge of the penalty area.
A second booking meant Blake was subsequently sent off but Swansea held on for only their third league win of the season.
Cameron Toshack's side remain bottom on 14 points, four points from safety.
Chelsea lie fifth on 24 points.