Trailing 1-0 at halftime at Anfield, Udinese stunned the five-time European champions with a second-half fightback sparked by Antonio Di Natale's well-taken equalizer and completed by Giovanni Pasquale's drilled finish that proved to be the winner.
Anzhi Makhachkala is level on four points with Udinese at the top of Group A thanks to a 2-0 win over Young Boys, courtesy of a brace by veteran Cameroon striker Samuel Eto'o.
Inter scored all its goals in the first half in a 3-1 win at Neftci of Azerbaijan to leave Andrea Stramaccioni's side level on four points in Group H with Rubin Kazan, which beat Partizan Belgrade 2-0.
Lazio heads Group J on four points following a 1-0 home win over Maribor of Slovenia but Tottenham is languishing in third place after conceding a late equalizer in a 1-1 draw at struggling Panathinaikos.
The only dampener on the night for Serie A clubs was Napoli's 3-0 loss at PSV Eindhoven that ceded top spot in Group F to Dnipro.
Elsewhere, Cristian Rodriguez grabbed an injury-time winner for Atletico Madrid in a 1-0 victory over Plzen, the defending champions' second straight triumph in Group B. French club Lyon also left it late to beat Hapoel Kiryat Shmona 4-3 in a thrilling match in Israel, with Gueida Fofana scoring a stoppage-time winner.
However, there was no end to the woes of Athletic Bilbao, last season's beaten finalist, which lost 3-1 to Sparta Prague to slump to the bottom of Group I. Bilbao is also in the relegation zone in the Spanish league.
In other results, France forward Loic Remy scored twice in Group C leader Marseille's 5-1 thrashing of AEL Limassol, fellow French club Bordeaux was sunk 3-0 at Newcastle and Hannover beat Levante 2-1 despite playing with 10 men for 80 minutes.
In another surprising result, unheralded Hungarian side Videoton swept past Sporting Lisbon 3-0 in Group G.
Liverpool came into the match against Udinese in buoyant mood after finally ending its winless run in the Premier League with a 5-2 demolition of Norwich on Saturday.
It started well, with Jonjo Shelvey putting the team ahead in the 23rd with a fine header.
But the evergreen Di Natale equalized 32 seconds after the break and after Sebastan Coates headed into his own net, Pasquale ran onto Di Natale's lay-off and thumped a low shot into the corner in the 72nd.
Substitute Luis Suarez pulled a goal back from a free kick but late pressure failed to produce an equalizer for Liverpool.
Inter warmed up for Sunday's city derby with AC Milan with a victory courtesy of goals by Philippe Coutinho, Joel Obi and Marko Livaja.
“Even though there were a lot of changes to the side, Inter put in a really solid first half,” said Stramaccioni. “This group of players is on the way up.”
Ederson's 62nd-minute winner left Lazio in control of a group containing Spurs, who took a first-half lead through Michael Dawson but failed to make their dominance count before Toche leveled in the 77th for Panathinaikos.
The Greek team was there for the taking after its poor start in the domestic league, which resulted in officials throwing captain Costas Katsouranis off the team on Wednesday.
“There is a sense of disappointment,” said Tottenham manager Andre Villas-Boas, who won the Europa League with Porto in 2011.
“In the second half, we made a lot of mistakes in passing and that let them in the game. But a point away from home is positive.”
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