The Hungarian outfit were making their first appearance in the group stages of the competition and travelled to Anfield with hopes of an upset.
An expected avalanche of goals was not the case however, as Liverpool wasted a host of chances and were at times restricted to speculative long-range efforts.
The goal from Kuyt arrived on the stroke of half-time as the frustrated home side neared the break without making their mark before the Dutchman netted a rebound from a Fernando Torres effort.
Liverpool go joint top of Group E along with Lyon, who defeated Fiorentina with an identical scoreline in the group's other game on Wednesday.
First HalfThe home side's first meaningful attempt came on 17 minutes as Torres executed a snapshot from around eight yards after Glen Johnson's low cross, just missing the target at the near post.
The referee was forced to get his notebook out soon after as first Marcell Fodor received a yellow for body checking Yossi Benayoun before Steven Gerrard needlessly joined him.
The Reds captain was too eager to get on with play after Liverpool were awarded a free-kick and subsequently received a yellow card for not waiting on the referee's whistle on 26 minutes.
The Hungarians offered their first real threat on goal on the half-hour mark as Peter Czvitkovics capitalised on a poor Jamie Carragher clearance which dropped kindly for him to half-volley on target only for Pepe Reina to tip over.
The game was a little stretched at this point as a Laszlo Bodnar free-kick from all of 40 yards forced Reina into action again before a flurry of chances for Albert Riera went begging.
Gerrard perhaps should have done better on 44 minutes as he steered a Riera cutback wide from just inside the area as the home side wasted another opportunity to take a half-time lead.
However, Kuyt eased the home side's nerves a minute into first-half stoppage-time as he reacted first to a Poleksic parry from a Torres effort and side-footed into the bottom corner from around 12 yards.
Second halfThe home crowd were appealing for a penalty on 51 minutes as Gerrard produced a brilliant nutmeg before losing his balance under a challenge from the left-back, only for the referee to wave away protests.
The Liverpool captain then went close minutes later with a spectacular volley from the edge of the area, narrowly evading the crossbar with Poleksic beaten.
Gerrard continued to show his eagerness to get on the score-sheet as he drilled in another long-range effort on the hour mark with the keeper comfortably saving this time.
Benayoun, who netted a hat-trick at the weekend, showed some fine silky wing-play on 65 minutes, beating four players before winning a corner with a well-struck shot from a tight angle.
Luis Ramos was replaced by Zsolt Laczko on 67 minutes as an injury brought the midfielder's evening to an end while Lucas headed wide unmarked from eight yards as Ramos received treatment on the sideline.
The warning signs were still there for Liverpool on 78 minutes as Adamo Coulibaly almost nipped in on goal only for Carragher to nod wide with Reina stranded.
The introduction of Ryan Babel sparked a little life into the home side and it was the winger who ignited an attack on 83 minutes as he played in Torres who cut the ball back across the face of goal where the on-running Benayoun blazed over with space in abundance.
The chance proved to be Benayoun's last as Javier Mascherano made a late entrance in a bid by boss Rafa Benitez to snuff out the away side's threat.
There was still time for another effort on goal from Gerrard as Babel teed up his captain on the edge of the area but the 29-year-old dragged his shot wide, summing up his night.
There was a late cameo appearance for Fabio Aurelio who saw out injury-time after replacing goal hero Kuyt as Liverpool held on for three points despite late pressure from the Hungarians.
LiverpoolTeam StatisticsDebreceni1Goals011st Half Goals07Shots on Target316Shots off Target82Blocked Shots510Corners69Fouls124Offsides31Yellow Cards10Red Cards081.3Passing Success71.228Tackles2982.1Tackles Success75.968.1Possession31.951.4Territorial Advantage48.6