The striker then had another opportunity when meeting Alonso's clipped cross from the right but goalkeeper Rame did enough to divert the ball into the side-netting from close range.
Liverpool's play, though, smacked of a team lacking in confidence, with passes failing to find their target and the defence unnecessarily panicking at times against decidedly average opposition.
And they were given a more stern examination at the start of the second half as Bordeaux increased the tempo, Fernando Menegazzo shooting at Reina from 18 yards.
However, Liverpool then went ahead on 58 minutes as Crouch made amends for his earlier miss. A fine diagonal pass from Riise was controlled by Garcia on the right, who cut inside Wendel before releasing a shot parried by the sprawling Rame. Gonzalez retrieved the loose ball and, after the Chilean's cross was put behind, Crouch was first to Bellamy's corner from the right to head in from six yards.
Within moments Crouch was replaced by Dirk Kuyt, the Dutchman having recovered sufficiently from the ankle injury sustained last week and now in with a chance of appearing at Old Trafford on Saturday.
The goal knocked Bordeaux out of their stride as Liverpool grew in confidence, although Carragher had to be alert to turn behind Julien Faubert's centre with Jean-Claude Darcheville waiting to pounce.
But with Sissoko also introduced from the bench for the final quarter, Liverpool sought to take a grip on proceedings and defend their advantage.
Bordeaux came close with eight minutes remaining when substitute Marouane Chamakh flashed inches wide from Wendel's free-kick on the left, before Menegazzo headed over a Frank Jurietti cross.
Stephen Warnock, on for Bellamy, could have sealed the win in the 89th minute but his shot was saved by Rame after the defender waltzed through with Bordeaux committing more players forward in search of an equaliser.
Liverpool, though, were not to be denied the victory Benitez will hope can kick-start their spluttering season.
BORDEAUX (4-4-2): Rame; Jurietti, Henrique, Jem-mali, Wendel; A Alonso (Faubert 63), Mavuba, Micoud, Fernando Menegazzo; Laslandes (Chamakh 63), Darcheville (Perea 71). Subs: Valverde, Ducasse, Enakarhire, Marange.
BOOKINGS: Jurietti (foul) and Rame (dissent).
LIVERPOOL (4-4-2): Reina; Finnan, Carragher, Hyypia, Riise; Garcia, X Alonso, Zenden, Gonzalez (Sissoko 69); Crouch (Kuyt 65), Bellamy (Warnock 87). Subs: Dudek, Peltier, Paletta, Pennant.
BOOKINGS: Zenden and Kuyt (both fouls).
REFEREE: Tom Henning Ovrebo (Norway).
ATT: 33,000.
NEXT GAME: Manchester United v Liverpool, Barclays Premiership, Sunday October 22, kick-off 1pm
