The game was settled in the 18th minute of a rip-roaring first half when Kuqi bravely dived to head a low right-wing cross from Craig Beattie past England Under-21 keeper Scott Loach, who got a hand to the ball but was unable to keep it out.
After that the superb defensive resilience of Paulo Sousa's men again shone through as they frustrated their hard-working opponents en route to keeping yet another clean sheet.
For the Hornets, who themselves looked capable of mounting a play-off challenge not so long ago, it was a third straight defeat and leaves them just two points clear of the drop zone, meaning Saturday's home game with Peterborough United now has the feel of a must-win to it.
The only surprise of a barnstorming end-to-end first half that yielded a total of 22 goal attempts was that it featured just the one goal.
Danny Graham saw a header cleared from near the line, while at the other end Beattie forced Loach into a good save and Kuqi headed against the post before the Crystal Palace striker scored what was to be the winner.
After that, the pick of Watford's chances in the opening half fell to skipper Jay DeMerit, who headed a corner narrowly over and then saw another effort from a set piece cleared off the line.
But Swansea had their chances too, as Loach excelled to thwart Beattie shortly before the forward fizzed another attempt narrowly wide.
The Hornets again had the majority of the possession after the break, but their end product was a lot more limited as Swansea switched to a five-man midfield, stifling their opponents' attempts to play through them.
Watford's best chance of an equaliser fell to Heidar Helguson, whose volley was well saved by Dorus DeVries, while substitute Guillem Bauza might have snatched a second goal on the counter attack for the visitors, had it not been for some fine covering defending from Lloyd Doyley.