Wilfried Bony had the chance to put the visitors in front within the first minute, but his effort was saved. Meanwhile, Djibril Cisse could have got on the scoresheet for the home side after he went one on one with goalkeeper Michel Vorm.
The striker scuppered his chance, allowing Swansea to find their rhythm, which they eventually did and rewarded themselves with a goal through Bony.
Some intricate passing and movement from the Swans found its way to Roland Lamah, who stepped around the last defender to square the ball to the left, which Bony latched on to and fired into the net from close range.
The Premier League outfit had a number of chances to extend their lead, but Kuban responded well to conceding by putting some passes of their own together.
The home side came close again, but Vorm was able to stop a shot and allow Swansea to go on the counter-attack, which reached Nathan Dyer, who curled a 25-yard effort towards goal. However, his shot was tipped away by the keeper.