Titleholders Atletico Madrid qualified with a 1-0 win over Hapoel Tel Aviv but last season's runners-up Athletic Bilbao, whose game in Israel against Hapoel Kiryat Shmona was postponed for security reasons, were knocked out without playing.
Sparta Prague, Levante, Genk, Lazio, Borussia Moenchengladbach, Fenerbahce, Viktoria Plzen, Dnipro Dnipropetrovsk, Bordeaux, Newcastle United and Napoli also went through.
Christopher Samba and Eto'o scored in a three-minute spell in the second half to steer Guus Hiddink's Anzhi team into the last 32.
Samba headed in from a corner and former Barcelona and Inter Milan forward Eto'o raced clear of the Udinese defence before slotting the ball past Zeljko Brkic, ending an unhappy European campaign for the Serie A side who missed out on the Champions League after losing on penalties to Braga.
The Group A match was played in Moscow after UEFA refused to let Anzhi stage matches in the violence-plagued Dagestan region for security reasons.
In the group's other game, Liverpool twice led but were held to a 2-2 draw by Young Boys after Montenegro's Elsad Zverotic struck for the Swiss visitors with a 25-metre rocket in the 88th minute.
Sparta Prague's 1-1 draw at home to Lyon in Group I sent them through alongside the already-qualified Ligue 1 outfit, but the result meant elimination for Marcelo Bielsa's Athletic.
Steve McLaren's Twente Enschede were also knocked out after being held to a scoreless draw by Hannover.
Elyaniv Barda's first-half header gave Genk a 1-0 win at Videoton as the Belgians progressed from Group G into the round of 32.
Napoli went through in dramatic style when Edinson Cavani, who scored four goals in their previous match against Dnipro, converted a penalty in the fourth minute of stoppage time for a 2-1 win at AIK after they had been reduced to 10 men.
Salvatore Aronica was sent off for the visitors for bringing down Mohamed Bangura as he had a run on goal.
Two first half goals from Brazilian Jussie led Bordeaux to a 2-1 win at Bruges in Group D while Newcastle's 1-1 draw at home to Maritimo in the same group was enough for them to progress.
Bekir Irtegun scored with a bicycle kick to give Fenerbahce a 1-0 win at Marseille in Group C, connecting acrobatically after a Cristian corner had been flicked on.
An early goal by Raul Garcia gave Atletico Madrid a 1-0 over Hapoel and a place in the last 32 and Czech side Viktoria Plzen followed them through from Group B by drawing 1-1 at Academica Coimbra.