The result was the all-important factor, as Liverpool knew they needed to equal or better the result gained by Young Boys against Anzhi Makhachkala, and as the Swiss side won 3-1 in the end, only a victory would do for the Reds.
Jordan Henderson got the only goal of the game to send Liverpool through, though they had to survive a couple of late scares to avoid conceding the equaliser.
The scoreline ensures the Reds will have continental football after the new year once more, giving them another trophy to aim for this season.
Here are six things we learned in the game.
Steven Gerrard was ill, Daniel Agger is being carefully managed through the Premier League fixture list and Lucas Leiva has only recently returned from a long-term injury, so all three first-team regulars were left back in Liverpool.
Beyond that, the entire first-team squad traveled to Italy for the crunch match against Udinese, and manager Brendan Rodgers showed his intent by naming a very strong starting XI including Pepe Reina, Martin Skrtel, Glen Johnson, Joe Allen and Luis Suarez.
One or two Europa League regulars also made the team, including Stewart Downing and Jordan Henderson, but the message was clear from the off from Rodgers—this was a competition Liverpool had every intention of staying in.
Long distance trips and a shortage of senior players have at times forced the boss' hand into playing inexperienced players, but in general, Rodgers has done a fine job of managing his squad through the group stage and coming out successful.
As this is his first campaign in Europe, he will be desperate to have some relative success, and the first stage has been well-negotiated in the end.
Europa League matches have seen chances come and go for Jordan Henderson this season.
The former Sunderland man has been played at right-back, been switched out of position only a few minutes into games and in general, been used as a utility player rather than getting his chance in his preferred position.
When he has gotten a run in midfield, he has on occasions gotten into dangerous positions in the box—but neglected to shoot, instead passing to teammates and seeing the chance lost. The game in Russia against Anzhi provided maybe the best example of this, but certainly not the only one.
This time against Udinese, he didn't hesitate, shooting home low to give Liverpool the lead, one which they never relinquished.
The Reds have needed goals from midfield all season, and just this week Brendan Rodgers has spoken about how his team need to pitch in with more strikes.
Henderson's goal was vital in that it sent the team through, but it was also pleasing in that it did not rely on an own goal or Luis Suarez doing the finishing work.
Liverpool will be looking to be active in the January transfer window, particularly with regard to bringing in attacking reinforcements.
The FA Cup will be there to play for from early 2013—the Reds travel to Mansfield or Lincoln in the third round—and of course, they'll be aiming for as high a league placing as possible, but every additional chance of winning silverware is a potential bonus.
Playing in European competition is important for every player, and although it's not the Champions League, at least Liverpool will be afforded more games and will travel to other countries as they continue their Europa League adventure.
Not every club in the North West of England can offer the lure of continental competition after the new year...
After Liverpool's first home game in the group ended in a surprise 3-2 defeat to Udinese, it seemed that the Reds had a lot of work to do just to qualify at all.
Three points against Anzhi at home, a defeat away and a sloppy draw at home to Young Boys, just made it even more difficult for fans to imagine anything other than second place being the target.
However, Liverpool's win in Udine coupled with a 3-1 Young Boys victory over Anzhi left the Russian and Swiss sides tied on points with the Reds—and Liverpool ran out group winners because of the head-to-head mini-league results.
Qualifying top of the group obviously comes with benefits; Liverpool will be seeded and therefore avoid all the sides who won their respective Europa League groups, as well as the four best third-placed teams who drop into the competition from the Champions League.
That is not to say, of course, that the round of 32 will be easy—far from it.
Liverpool can face any second place team, and that includes the likes of Atletico Madrid, the current holders, and the impressive young German side Borussia Mönchengladbach.
A tie with the Bundesliga side would of course evoke memories of European glories gone by for Liverpool.
The last 10 minutes of the tie against Udinese were tense and worrisome for Liverpool.
A needless and ridiculous late concession of a free kick could have been extremely costly, as just one late goal from Udinese would have sent Liverpool out of the competition entirely with seconds to go of injury time.
Antonio Di Natale fired over from inside the box with less than 10 seconds of the three added minutes left to play.
The Reds did survive, however, and take their place in the next round, showing that the end result is all important and that any near-misses are meaningless, but ultimately, the club shouldn't have put themselves in the position that they were anxious in the final stages of the game.
Luis Suarez had at least two great opportunities to score the second goal, Suso missed a real sitter and Jordan Henderson saw a shot saved too.
The goals will come eventually, and the best thing that can be said is that the recent improvement in defence for the Reds has ensured that one goal has been enough.
A win is a win, and consecutive 1-0 victories over Southampton and now Udinese have proved very important for Liverpool.
It has been 11 years since Liverpool last won Europe's second competition—the UEFA Cup as it was at the time when Jari Litmanen, Sami Hyypia and Robbie Fowler lifted the trophy in Dortmund.
Since then, Liverpool have been to two European Cup finals and almost reached another Europa League final too, but could they go on to lift the trophy this year?
The bloated Europa League format means that there is still an awfully long way to go before the Reds can even begin to think about their chances for success. The final is still nine games away, and eight very good sides are about to enter the competition from the Champions League.
Other group winners alongside Liverpool include Fenerbahce, Dnipro Dnipropetrovsk and Lyon, all of whom are performing very well domestically.
It will be a tough competition to win, as always, and the fact of the matter remains that until the quarterfinals or so, not all teams may play their strongest XI.
Liverpool have a chance, and they are still in the competition. At this stage, that would have been Rodgers' and the team's only real objective.
January will bring the club a chance to improve and strengthen—and then a trophy and a good run in the Premier League will certainly not be beyond Liverpool.