Antonio Di Natale starred for Udinese as the Italian team edged past Liverpool 3-2, with Newcastle, Inter Milan, Lazio and Atletico Madrid also winning their Europa League matches.
Last year's finalists Athletic Bilbao surprisingly went down to Sparta Prague 3-1, while Marseille and PSV Eindhoven picked up impressive victories. Tottenham Hotspur had to settle for a point after Panathinaikos netted an equalizer 20 minutes from time.
At Anfield, Liverpool started the better side, with Oussama Assaidi, Jonjo Shelvey and Sebastian Coates seeing their efforts thwarted. The home side, however, took the lead in the 23rd minute, after surviving a couple of scares from the excellent Di Natale, with Shelvey starting the move and finishing it.
The youngster picked up the ball before passing it wide to Stewart Downing, who saw his cross powerfully headed home by Shelvey, giving goalkeeper Zeljko Brkic no chance.
Liverpool looked comfortable after the goal, with the only threat again coming from the Italian international Di Natale, who saw a strike go just beyond the crossbar. Brendan Rodgers' side were in control, but the home team failed to find the second goal, despite their dominance.
Liverpool paid for their misses early in the second half, when Glen Johnson gave the ball away dangerously, allowing Di Natale to pounce and play a one-two with Andrea Lazzari, before clinically slotting the ball past Pepe Reina.
Rodgers brought on captain Steven Gerrard and last game's hat-trick hero Luis Suarez, but it was Udinese that scored the next two goals.
Coates, unluckily, headed the ball into his own net off a free kick in the 70th minute, before Di Natale set up Antonio Pasquale for the third goal, two minutes later.
Liverpool reduced the deficit through Suarez, who curled in a wonderful free kick, but despite some strong efforts in the final moments, Udinese held on for the win.
At St James' Park, there was no such drama, as Newcastle comfortable held off the challenge of French side Bordeaux, with Shola Ameobi opening the scoring in the 16th minute. It was 2-0 five minutes from half time, with defender Henrique putting the ball into his own net, before Papiss Cisse scored a brilliant goal early in the second half to ensure all three points for Alan Pardew's side.
Results: Neftci 1-3 Internazionale; Anzhii 2-0 Young Boys; Rubin 2-0 Partizan; Panathinaikos 1-1 Tottenham; Metalist 2-0 Rapid Wien; Basel 2-2 Genk; Videoton 3-0 Sporting; H. Kiryat Shmona 3-4 Lyon; Sparta Praha 3-1 Athletic; Lazio 1-0 Maribor; Rosenborg 0-1 Leverkusen; Hannover 2-1 Levante; Helsingborg 2-2 Twente; Atlético 1-0 Plzen; Marseille 5-1 AEL; Mönchengladbach 2-4 Fenerbahce; Club Brugge 2-0 Marítimo; Molde 2-0 Stuttgart; AIK 2-3 Dnipro; PSV 3-0 Napoli; Steaua 1-0 FC Copenhagen; Liverpool 2-3 Udinese; Académica 1-1 H. Tel-Aviv; Newcastle 3-0 Bordeaux.