The Finland international finished last season as the Eagles' top-scorer but got a mixed reception from home fans when his name was announced before kick-off.
But Kuqi, in his first full start for the Welsh side, refused to celebrate when he slid the ball past the advancing Julian Speroni in the 14th minute, opening his goal account for Paulo Sousa's side after a January move from German outfit Koblenz.
It was enough to score a first league victory for Swansea at Selhurst Park since September 1948 and added further credibility to their hopes of plying their trade in the Premier League next season.
They were clearly the better side in the opening 45 minutes against a strangely subdued home side, who did not play with the swagger expected after winning their previous three games.
Nathan Dyer fired into the side-netting just before the half hour mark but clear chances were few and far between.
Swansea went close to extending their lead shortly after the break when Andrea Orlandi played a one-two with Adam Tate to fashion a shooting chance inside the box but his low shot was pushed away for a corner by Speroni.
Palace substitute Kieran Djilali produced a wild finish over the top when he tried to finish off a flick on from Calvin Andrew on the hour mark.
Eagles captain Shaun Derry looked lucky to stay on the pitch after a two-footed tackle on Kuqi in the 72nd minute and Sousa was equally incensed about a similar challenge by Andrew on Darren Pratley.
Palace's best chance of an equaliser was wasted in the 74th minute. Johnny Ertl's free-kick was glanced to the back post by Clint Hill but top-scorer Darren Ambrose volleyed across the face of goal.