The experienced frontman was offered a new deal to stay at Palace in the summer but was not happy with the terms and opted to leave the troubled London club.
And Kuqi showed cash-strapped Palace exactly what they were missing as he tucked home a neat first-half finish to keep the Swans' promotion bid on track and end Neil Warnock's side's run of three consecutive victories.
Swansea started to dictate the play early on with their trademark slick passing and movement and Spanish midfielder Andrea Orlandi tried his luck from long range but Julian Speroni made a comfortable save.
And it came as no surprise when the visitors took the lead after 14 minutes courtesy of the Kuqi.
Orlandi was the architect, threading the ball through to the Finland international who kept his composure and tucked a neat finish beyond Speroni from an acute angle.
Palace responded and started to compete in midfield but Swansea almost caught them out 28 minutes in when winger Nathan Dyer nipped in but dragged his shot into the side-netting from 12 yards out.
The Eagles felt they should have had a penalty 12 minutes before half-time when Alan Lee went down under a challenge from Kuqi, but referee Graham Scott saw nothing untoward.
Lee was causing some real problems for the Swans' defence but the experienced striker failed to get a strong connection on a cross from Lee Hills.
Paulo Sousa's side were almost two goals to the good four minutes into the second half but Argentinian keeper Speroni made a magnificent save to deny the lively Orlandi.
Defender Kieran Djilali was presented with an opportunity to draw the home side level but his finish looped over Dorus De Vries and over the crossbar.
Palace showed no shortage of endeavour but they were finding it difficult to break down a stubborn Swansea side and De Vries was rarely called into action.
Swans full-back Fede Bessone let fly with a long-range drive but his effort was never going to trouble Speroni.
Palace continued to push forward in search of the equaliser and Sean Scannell saw his overhead kick land just over the bar from close range but that was as close as Palace came.