So it was in extremely cool conditions that the game began but the visitors started brightly and Darlington goalkeeper Sam Russell was called into action in the fourth minute when saving from Lee Trundle.
Darlington quickly responded and a low drive from Mark Convery went narrowly wide before striker Alun Armstrong dived full length to connect with a David McGurk cross but his effort went wide.
Convery was involved again when the right winger's cross found the head of Clyde Wijnhard, but this time it was Swansea keeper Willy Gueret who saved.
Darlington took the lead in the 34th minute when a Bobby Petta free-kick was glanced home by the head of defender Matthew Clarke.
The visitors were quickly back on level terms and in the 36th minute Adrian Forbes was given too much space inside the area and he easily converted a left-wing cross from Izzy Iriekpen.
The start of the second half came under threat after a heavy fall of snow at the break covered the pitch in a blanket of white.
Referee Colin Webster allowed the game to continue and the visitors should have taken the lead in the 49th minute.
A cross-field pass was intercepted by Kevin McLoad who broke clear and looked certain to score until Sam Russell pulled off a magnificent save.
In the 65th minute Clarke headed over the crossbar when meeting a Petta corner and a long-range effort from Wijnhard went narrowly wide.
The Swansea defence was under threat and they finally fell in the 84th minute when Neil Wainwright struck the winner from close range after the defence had failed to clear a free-kick from Petta.