The Premier League club were joined by compatriots Tottenham Hotspur, former European champions Steaua Bucharest, VfB Stuttgart and Basel as the last five clubs through to the knockout stages.
Inter Milan, already qualified, were astonishingly held to a 2-2 draw at home by Azerbaijan's Nefti Baku.
Liverpool midfielder Jordan Henderson's first strike of the season after 23 minutes helped Liverpool to leapfrog Group A leaders Anzhi Makhachkala, though Antonio Di Natale blazed a good chance over the bar with the last kick of the game.
Udinese played the last 11 minutes with 10 men after Giovanni Pasquale's dismissal and Di Natale could have eliminated Liverpool altogether but he wasted a great late chance as Anzhi lost 3-1 at Young Boys in Berne.
Liverpool, Anzhi and Young Boys all finished on 10 points but the Russians' better record over the Swiss club in a three-way head-to-head took them into the knockout stages along with the English group winners.
"We were in a difficult group with good teams," said Liverpool manager Brendan Rogers. "But the mentality of the team was very strong and our desire and belief was very strong. We won and that is good for our future."
Steaua's 10 men held Copenhagen 1-1 to progress as Group E winners on 11 points with Stuttgart in second on eight and level with the Danes but the Germans progressed owing to a better head-to-head record.
Basel's 0-0 draw at Group G's top club Racing Genk ensured that Hungary's Fehervar, whose match at Sporting was postponed to Friday due to a waterlogged pitch, went out.
In Group C, Fenerbahce had already made sure of top place ahead of Borussia Monchengladbach while in Group F Ukraine's Dnipro Dnipropetrovsk, 4-0 winners over AIK Solna, had secured first place ahead of Napoli, who lost 3-1 at home to PSV Eindhoven b
