Liverpool were not at their best against the Hungarian champions and will need to improve if they are to advance from Group E which also contains Fiorentina and Olympique Lyon.
Only a fine one-handed save by Jose Reina prevented Peter Czvitkovics embarrassing Liverpool in the 30th minute.
Fifteen minutes later Kuyt slid in to score from the rebound after keeper Vukasin Poleksic had stopped a shot by Fernando Torres.
"We had some chances and didn't take them," Liverpool manager Rafa Benitez told reporters. "We put ourselves under pressure at the end, we couldn't score a second goal.
"Hopefully we will see better things in the next game (at Fiorentina on September 29). The most important thing is to start the Champions League group stage with three points."
Liverpool began the match slowly before upping the tempo towards the end of the first period, Poleksic producing two saves in as many minutes to frustrate Albert Riera before Kuyt netted.
Home captain Steven Gerrard, who was earlier booked for taking a free kick too quickly, was guilty of the miss of the match.
The England international was 12 metres from goal and unmarked but scuffed his shot wide of the post.
Gerrard also went close with a right-foot volley which went narrowly over the crossbar.
Debrecen played well and created several chances without finding the finishing touch.
Coach Andras Herczeg said: "We did our best and our performance gives us confidence for our next match (at home to Lyon)."
(Editing by Tony Jimenez)