A barely merited 1-0 win over Bordeaux on an evening when Benítez's adventurous tactics might easily have backfired on him will hardly have imbued Liverpool's supporters with confidence before their game away to Manchester United in the Barclays Premiership on Sunday, but it leaves Benítez's side well placed to progress to the Champions League knockout stages.
With Liverpool's title challenge hinging on the outcome of their trip to Old Trafford, too, their first away win of the season was perfectly timed, although there was little on show to give Sir Alex Ferguson, the United manager, much to fear.
Crouch's 58th-minute header proved decisive, but Bordeaux — while never really carving out gilt-edged chances — had their moments, not least when Marouane Chamakh nodded wide when well placed with nine minutes remaining.
Stephen Warnock might have added a little gloss to the scoreline at the death when he was denied from a one-on-one chance, but that would have been unduly harsh on Bordeaux.
If the game did anything, it served to emphasise how important Steven Gerrard and Mohamed Sissoko are to the Liverpool cause, although with the former sidelined with a hamstring injury, and doubtful for the United game, it was no surprise to see Benítez start with Sissoko on the substitutes' bench rather than risk aggravating the ankle problem that kept him out of the 1-1 draw against Blackburn Rovers at the weekend.
In their place, Boudewijn Zenden partnered Xabi Alonso in central midfield, but Benítez would not be Benítez if he did not spring one or two surprises. Still, the inclusion of Mark González on the left wing in addition to Luis García on the opposite flank was unexpected, even by the manager's unpredictable standards.
Benítez had talked about the need for improving defensively, especially away from home, but not even a clean sheet could mask the susceptibility from wide areas that has been Liverpool's chief failing this season.
Given time and space, Alejandro César Alonso and Fernando Menegazzo needed little prompting to raid forward as García and González routinely left Steve Finnan and John Arne Riise, the full backs, isolated and exposed.
Liverpool were lucky not to fall behind in the 25th minute. Wendell's whipped free kick seemed there for the taking, but with Finnan stood around aimlessly, Johan Micoud fluffed his header.
Crouch was not enjoying the best of nights up until his goal and when, unmarked from six yards out, he volleyed Craig Bellamy's delightful looped ball wide, the England forward looked more like the man who went four months without scoring for Liverpool than the player who had plundered ten goals in his previous 15 games for club and country.
Presented with another inviting chance four minutes later, Crouch was more composed, cushioning Alonso's raking pass, evading Franck Jurietti's challenge, and thwarted only by the legs of Ulrich Ramé.
Despite the danger Bordeaux continued to pose on the counter-attack, Liverpool always looked capable of sneaking a goal and so it proved.
Crouch may have been subdued, but he could never be accused of hiding, and in powering home Bellamy's corner with a fine header after the interval, the beanpole underlined his character as much as his burgeoning credentials as a goalscorer. García might have scored before then, but, cutting inside his marker, the Spaniard's venomous shot was palmed to safety by Ramé.
As Bordeaux continued to get behind Liverpool's full backs and pummel balls into the box, Carragher and Hyypia remained heroic to the last.
Menegazzo squandered an inviting opening shortly after the restart when he fired straight at José Manuel Reina after the ball had deflected kindly to him, but the longer the game went on, the more frustrated Bordeaux grew.
González's departure for Sissoko with 20 minutes remaining drew a welcome sigh of relief from Liverpool's fans and, no doubt, Riise, and when the final whistle sounded, the overwhelming feeling was, as Carragher later articulated, of a “job done”, if not at all convincingly.