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.
"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 Sept. 29). The most important thing is to start the Champions League group stage with three points."
Coach Andras Herczeg said: "We did our best and our performance gives us confidence for our next match (at home to Lyon)."
