It was sufficient to leave them favourites to win Group C, while effectively ending Bordeaux's hopes of an extended Champions League campaign. But Rafa Benítez's side missed Steven Gerrard's customary midfield dynamism and will need to be far more inventive if their flagging Premiership campaign is to be reignited at Old Trafford on Sunday.
After a series of underwhelming league performances, a team meeting billed as a crisis summit and his demands that Liverpool's senior professionals start pulling their weight, Benítez looked a relieved man at the final whistle when he indulged in unusually extravagant celebrations.
Peter Crouch's header from Craig Bellamy's corner had secured his team's first away win of the season and the manager admitted: "It was important to win away and get a clean sheet in terms of the players' confidence. It was difficult but we worked really hard to control the game and my players deserve a lot of credit."
With, depending on your degree of cynicism, Gerrard either hamstrung or rested before the date with Manchester United, Bolo Zenden was offered a rare start in central midfield. Starting slowly, he was swiftly dispossessed by Lilian Laslandes. The Bordeaux striker's lumbering gait was familiar to anyone who watched him during his woeful stint at Sunderland five years ago, but a series of intelligent off-the-ball runs and impressive interceptions must have come as an irritation for Liverpool who, initially at least, found the 35-year-old a nuisance.
Shortly after Steve Finnan was required to make an important headed defensive clearance, Wendel whipped in a free-kick which found Johan Micoud. Fortunately for Benítez's central defensive pairing, Micoud failed to make proper headed contact and a glorious chance went wide.
Nevertheless, much to the disappointment of those home fans clutching banners emblazoned with "Bordeaux Piranhas", the French attack lacked bite and Pepe Reina was largely redundant on a night when, with Jerzy Dudek, his understudy, facing a potential five-match suspension for his part in a rumpus in a reserve game against Everton, Benítez must have been praying he avoided injury.
Yet worryingly for Liverpool's manager the Merseysiders struggled to conjure even half-chances from open play and much of that was down to Rio Mavuba. Born at sea on an overcrowded, rickety boat making the hazardous journey from Angola to France, Mavuba is evidently not a man to be daunted by expensive reputations.
Bordeaux's backline proved as capable. Tellingly, on the odd occasion Bellamy's pace seemed to have offered him an escape route from Henrique, he was invariably flagged offside.
Jean-Claude Darcheville might have done better with a chance early in the second period, that opening having originated in good work from Fernando Menegazzo, whose impressive contribution was lending Bordeaux an assurance which, for periods, Liverpool appeared unable to ruffle.
Similarly, Mark González was making a negligible left-wing impact while, across on the right, Luis García struggled until he finally managed to burst beyond the left-back and dispatch an angled shot which Ulrich Ramé tipped away for a corner.
That dead ball was, perhaps surprisingly, executed by Bellamy, whose delivery was met by Crouch's head and instantly redirected into the back of the net.
Bordeaux, though, were not quite done and Jamie Carragher, captaining Liverpool in Gerrard's absence, needed to be back at something approaching his best when clearing a dangerous cross supplied by the substitute Julien Faubert.
With Faubert proving an awkward attacking customer, Carragher and Sami Hyypia were offered late scope to prove they are not in terminal decline after all but Sir Alex Ferguson will doubtless have noted that Benítez's backline has not entirely eradicated its new-found nervousness in the face of decent crosses.