Title-holders 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 Monchengladbach, Fenerbahce, Viktoria Plzen, Dnipro Dnipropetrovsk, Girondins Bordeaux, Newcastle United and Napoli also went through.
Lazio's goalless draw at home to Tottenham Hotspur was played against a backdrop of violence after 10 supporters of the English side were injured in an attack on a Rome pub by dozens of masked men in the early hours of Thursday.
Police said that among 15 Italians detained were two fans of AS Roma, Lazio's bitter rivals, who were charged with violence and grievous bodily harm.
"It was not easy, we struggled against a good Lazio team. They controlled the game, they were well-organised and we had trouble to find space, so it is a good point for us," he told French TV.
"I may have had much more work than usual in the Europa League, it is good for me and it is good for the team to grab a point."
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.
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 96.
