"The most important thing is to get the three points," Reds manager Rafael Benitez said after watching his side overcome the Hungarian champions thanks to a goal from Dutch striker Kuyt in added time at the end of the first half.
"People didn't know what to expect because they didn't know the other team, but they showed they're not bad," added the Spaniard who was celebrating his 300th game in charge of Liverpool.
Debrecen, playing their first-ever tie in the Champions League group stages, succeeded in thwarting the attacking threat posed by Steven Gerrard, Fernando Torres and Kuyt.
"We had three or four good chances and if we take our chances it would be easier," said Benitez.
In the other Group E tie, French club Lyon defeated Fiorentina of Italy 1-0 at home.
Benitez began the match in Liverpool with the same side that had defeated Burnley 4-0 on Saturday which meant Argentina midfielder Javier Mascherano started on the bench following a short spell out through injury.
Israeli Yossi Benayoun, scorer of a hat-trick against Burnley, was involved in creating Liverpool's only goal against Debrecen before making way for Mascherano near the end.
Debrecen, backed by a few hundred noisy fans, defended solidly early on, restricting the home side to long-range strikes fired tamely wide of the visitors' goal.
The Hungarians grew in confidence as the first-half progressed and in the 30th minute Peter Czvitkovics forced Pepe Reina in the Liverpool goal into a finger-tip save over his bar.
It was a wake-up call for Benitez's players who fought back thanks to two strikes from Albert Riera but on each occasion the Spaniard's effort was well saved by Debrecen's goalkeeper Vukasin Poleksic.
Riera was at the heart of the action once more just before the break, firing in a low cross but Liverpool captain Gerrard was unable to keep his shot on target.
Liverpool finally opened the scoring moments before the end of the first half.
Torres, collecting a through-pass from Benayoun, swivelled away from his marker on the edge of the Debrecen box, before unleashing a shot towards goal. Poleksic managed only to push the ball back into the path of Kuyt and the Dutchman slotted home from close range.
Liverpool took the game to Debrecen in the second half as Gerrard shot a volley just over the bar and a Benayoun effort was punched away by the keeper after a mazy run from the diminutive midfielder.
Debrecen might have snatched a shock equaliser near the end but French striker Adamo Coulibaly, finding himself free in the box after beating the offside trap, was unable to lob Reina rushing from his goal.