Although David Ngog scored a fourth-minute goal for the English Premier League side, Liverpool were eliminated after Fiorentina defeated Lyon in the other group game.
Liverpool got off to a flying start with some assistance from fallible defending when David Ngog poked in Jamie Carragher's header from a corner
Debrecen had their best chance 10 minutes from time but Gergely Rudolf's sharp shot from a narrow angle was parried wide by Pepe Reina.
Liverpool, who failed for the first time to reach the knockout stages under manager Rafa Benitez, must now settle for a place in the Europa league.
"You have to be disappointed, we did our job today, but we are out. That is something we cannot change now. We made mistakes," Benitez told reporters.
"Our next goal is to do well in the Europa League."
Captain Steven Gerrard said it had been disappointing to rely on other teams for the opportunity to progress.
"We've played fantastically well tonight and deserved the three points, so it's disappointing that we won't progress in the Champions League," he told the club website.
"You get what you deserve in this competition and the two late goals we conceded against Lyon have been crucial. But it's gone now, it's in the past and we've got to move on.
"Tonight's performance was encouraging and we can build on that. We've got a lot of players coming back from injury so hopefully it's onwards and upwards from here."
Budapest manager Andras Herczeg said: "We made a terrible mistake before the goal, but that is football.
"We are proud of ourselves that made life hard for teams of a calibre like Liverpool, Lyon and Fiorentina. We are desperate to get a point at the last match away to Lyon."
