The Ligue 1 side broke the deadlock shortly after the half-hour mark when Henri Saivet curled home a free kick from inside the box after Simon Mignolet had been penalised for holding on to it for too long.
That lead last just five minutes before James Milner tucked away a penalty after Christian Benteke had been fouled inside the box, while Benteke got his own name on the scoresheet right on the stroke of half time to complete the turnaround.
Here, Sports Mole takes an in-depth look at whether the hosts deserved to come away with the points on Merseyside.
On target: 4
On target: 2
Was the result fair?
This was not the best match to watch, but Liverpool just about edged it overall and they deserve their victory, although anything more than a one-goal margin may have flattered them a little. The first half was more even than expected as Bordeaux refused to sit back and try to nick one on the break, instead committing men forward and asking questions of the Liverpool defence. The opening goal did come against the run of play, though, and it was a rather bizarre way for a goal to come about as Mignolet was penalised on the six-second rule.
Liverpool had created chances through Roberto Firmino and Benteke before that, but it was important that they levelled things up soon after going behind. They too had a helping hand with their first goal of the game as it looked to be a very soft decision from referee Alon Yefet to award the penalty. Even so, Milner stuck it away, and Liverpool completed the turnaround with the last action of the first half as Benteke hit a fine volley on the turn past a helpless Cedric Carrasso.
Scoring just before half time is always an added bonus, and Liverpool certainly came out as the better side in the second. Benteke and Jordon Ibe both had big chances in the opening 10 minutes of the half, while the former also saw a goal harshly disallowed. That 10-minute spell was the only real time either side were very much on top throughout, though, and while Liverpool could have put the game to bed in that period, they were also a little fortunate not to be punished by a few Bordeaux chances late on.
Liverpool's display against Manchester City on Saturday justifiably got plenty of tongues wagging, and while it is of course unrealistic to expect that level of performance in every match, fans would have been keen to see at least some of the momentum spill over to tonight. The Reds have not exactly set many pulses racing at Anfield this season, and it was the same old story tonight as they put in a rather tepid display during what was a flat night on Merseyside, despite the result.
Firmino, one of the stars at the Etihad, was unable to replicate that form this evening, and Philippe Coutinho's absence left Liverpool a bit short in the creative ranks. They did carve out some chances, and probably should have scored more than the two they managed, but the general level of play was relatively uninspiring, particularly compared to what had just been. Defensively they also gifted the opposition another goal, which has been the case far too often this season.
The hosts did enjoy a good spell at the start of the second half, but aside from that they struggled to gain any sustained period of control over the game this evening. To have managed less possession and fewer shots than a relatively poor Bordeaux side at home is not really good enough, and vast improvement will be needed if they hope to go much further in this competition. Still, a win was what they required to qualify, and a win is what they got, so Jurgen Klopp will be content this evening.
The visitors came into this match with their Europa League hopes hanging in the balance and knowing that only a win was really good enough to keep their chances of reaching the knockout rounds alive. Even so, they took a few people by surprise at how readily they committed men forward, both on and off the ball. They were quick to press Liverpool's defenders high up the field and also enjoyed plenty of possession inside the opposition half, when some had been expecting them to play mainly on the counter-attack.
Bordeaux almost took the lead after just 13 minutes when Kolo Toure blocked Clement Chantome's effort after a Mignolet error, and it was another mistake from the Liverpool keeper that eventually did hand the visitors an opener. For all the time Willy Sagnol would have spent planning for this match, he would not have expected his side to be awarded an indirect free kick inside the box for Mignolet holding on to the ball for too long. Those situations are rarely as easy to score from as they appear, however, and Saivet did really well to find the top corner with his effort.
Conceding so soon after taking the lead was a major blow for the visitors, and they have every right to feel aggrieved about the decision from the referee. It could be argued that Ludovic Sane didn't need to make any contact with Benteke, but it was still a soft spot kick to give. There wasn't much they could do about Liverpool's second, though, and from then on Bordeaux's threat was minimal. They did end the game fairly strongly with a few chances, but ultimately they didn't do enough tonight and can have no complaints at crashing out of the tournament.
Sports Mole's man of the match
Christian Benteke: The vast majority of Liverpool's best moments this evening went through Benteke, who had the ball in the back of the net three times. Only one of those counted, with another harshly disallowed, but that proved to be the winner after the Belgian had already won the dubious penalty for his side. His performance will be all the more welcome after the news that Daniel Sturridge has picked up yet another injury.
This isn't a difficult one. It is incredibly rare to see a goalkeeper penalised for holding on to the ball for so long, but Mignolet didn't really give the referee much choice. The official rule is six seconds, and Mignolet had it in his hands for more than 21 before the free kick was awarded. It cost his side a goal, and the most peculiar thing about it is that there was no need to do it. It was far too early to be time wasting, particularly as the teams were level and Liverpool needed a win to progress.
Yefet did make that bold decision, but on the whole he had a poor game tonight. The sound of his whistle pierced the Anfield air all too often, including to award a soft penalty against Sane that gave Liverpool a route back into the match.
Liverpool: Next up for Klopp's Liverpool side is a match against Swansea City at Anfield on Sunday.
Bordeaux: Bordeaux, meanwhile, host high-flying Caen in Ligue 1, also on Sunday.