The Holland striker stabbed home in first-half injury time after goalkeeper Vukasin Poleksic half-saved Fernando Torres' shot.
Steven Gerrard flashed a volley over the crossbar midway through the second half but it was a frustrating night against the Hungarian champions, making their debut in the group stages.
The hosts were sluggish and uninspired and it took them an awfully long time in the first period to get some reward for their total domination.
In fact there were barely a couple of seconds left in the half to restart the match when Kuyt forced the ball home after Vukasin Poleksic had palmed out a Fernando Torres drive.
Before that it had been one-way traffic towards the Hungarian champions' goal. The nearest Liverpool came before Kuyt's strike was after 30 minutes when Norbert Meszaros kicked off the line from the Dutchman after another Poleksic save from Albert Riera.
The only time Jose Reina was called into action was to touch over a 20-yard effort from Meszaros.
Liverpool improved after the break, which was not difficult, and Gerrard smashed a swerving 25-yarder against the crossbar before Yossi Benayoun went on a brilliant run past a string of defenders before forcing Poleksic into a fumbling save at the foot of the near post.
Ryan Babel came on with 10 minutes left for Riera, his first appearance since yet another outburst over his lack of matches, and he produced one clever juggling act to fire just wide.
But Adamo Coulibaly, Debrecen's French frontman, almost punished Liverpool by forcing his way through and stabbing a shot wide after 82 minutes.