David Ngog saved Liverpool the blushes as they beat Unirea Urziceni of Romania 1-0 to register their first win at the Europa League campaign on Thursday.
The five-time European champions dropped down to the second-tier competition after their elimination at the group phase of the Champions League.
Ngog secured the breakthrough in the 81st minute giving them a slight advantage in the home leg on February 25.
The Frenchman scored with a close-range diving header after Ryan Babel's raking cross had been knocked down by 19-year-old substitute Daniel Pacheco, to hand Rafael Benitez's side a slim advantage.
"I think we'll get more joy away from home against them," said Liverpool captain Steven Gerrard.
"They'll have to come out and try to beat us. I'm confident we can win more convincingly away from home," he added.
Reigning UEFA Cup holders Shakhtar Donetsk went down to an unexpected 2-1 defeat by an impressive Fulham, after England hopeful Bobby Zamora scored a sensational winner with a powerful shot into the top-right corner in the 63rd minute.
"We got to grips with them a bit better in the second half and pushed them into a few errors. It was a great goal from Bobby which took the pressure off us defensively," said the Londoners' midfielder Danny Murphy.
On the other side Roma too crashed to a 3-2 loss to Greek side Panathinaikos. The hosts trailed twice but levelled in the 84th minute through Lazaros Hristodoulopoulos before former Liverpool striker Djibril Cisse fired in the winner a minute from time.
Elsewhere, a brace from Brazilian striker Amauri earned Juventus a precious 2-1 lead over Ajax following the first leg of their tie in Amsterdam.
Amauri struck either side of half-time in response to Miralem Sulejmani's 16th-minute opener to give Alberto Zaccheroni victory in his first European game in charge since replacing the sacked Ciro Ferrara.
Last season's runners-up Werder Bremen lost 1-0 in the away leg of their tie against FC Twente, who prevailed through Theo Janssen's 39th-minute goal.
Misfiring German champions Wolfsburg, 23 points off the pace in the defence of their Bundesliga title, drew 2-2 at Villarreal, while Belgian outfit Club Brugge secured a famous 1-0 victory against 10-man Valencia.
Hamburg defeated Dutch heavyweights PSV Eindhoven 1-0, while Marseille won 3-1 at FC Copenhagen and Atletico Madrid were held to a 1-1 draw at home to Turkish side Galatasaray.
Russian champions Rubin Kazan planted one foot in the last 16 after Aleksandr Bukharov scored a well-taken brace in a 3-0 win at home to Israel's Hapoel Tel-Aviv, with high-flying French side Lille edging Fenerbahce 2-1 in northern France.
Standard Liege of Belgium mounted a stirring comeback at home to Red Bull Salzburg, fighting back from 2-0 down at half-time to win 3-2 thanks to second-half strikes from Igor De Camargo and an Axel Witsel double.