The home side, managed by former Swans captain Dave Penney, took the lead after only two minutes when ex-Sunderland striker Martin Smith raced into the area before firing low into the back of the net beyond French keeper Willy Gueret.
The home side enjoyed the greater possession in the opening stages of the game and Swans manager Kenny Jackett was forced into a change when taking off defender Thomas Butler in the 13th minute.
Striker Adebayo Akinfenwa came on and two minutes later the visitors drew level.
Surprisingly it was the smallest player on the pitch who scored when a deep cross found midfielder Leon Britton at the far post and he beat Sam Russell from eight yards out.
The visitors almost took the lead in the 17th minute through Shaun MacDonald, but Russell saved well at full stretch.
In the 23rd minute Darlington defender Craig James fired a 20-yard free-kick inches wide and then in the 32nd minute, as the home side continued to press, Neil Wainwright drove an effort narrowly wide.
Further chances fell to the hosts and Alun Armstrong wasted a good opportunity when heading over in the 34th minute and then five minutes later the crossbar came to Swansea's rescue, when Michael Cummins fired against the woodwork for the home side.
But Darlington appeared to run out of steam as the half came to a close and in the 42nd minute Swans top scorer Lee Trundle fired into the side-netting from a good position.
The visitors began the second half as they had ended the first and they took the lead in the 51st minute when captain Darren Pratley's shot was turned away by Russell, but only into the path of the on-rushing Andy Robinson, who lashed the loose ball into the unguarded net from ten yards out.
Swansea continued to dominate the game and in the 77th minute Trundle again missed a great opportunity when firing across the face of goal.
However, just two minutes later the game was finally settled when Akinfenwa saw that Russell was off his goal-line and he fired a low drive into the far corner of the net.