Gerrard hastily snatched at a clear opening on 56 minutes after Crouch's knockdown had split the Bordeaux defence, one of few occasions Liverpool exploited the striker's aerial prowess of which the French defence had been so fearful.
And an increasingly tetchy game boiled over midway through the second half when the visitors were reduced to 10 men. Pierre Ducasse fell to the turf theatrically after a fair challenge from Mohamed Sissoko and Bordeaux kicked the ball out of play.
Riise, assuming there was nothing wrong with Ducasse, took a quick throw-in which Menegazzo took exception to and butted the left-back, leaving the Norwegian with blood pouring from a gash above his eye and his Brazilian assailant with a red card.
A man and a goal up, Liverpool pressed forward but Gerrard's woes continued moments later when he skied another decent chance well over from Finnan's ball over the top before finally getting his goal 18 minutes from time.
Four minutes later, Garcia made the game and Liverpool's progression to the knockout stages safe by dispossessing Bordeaux centre-back David Jemmali 35 yards and racing in on goal before knocking home beyond the helpless Rame.
It was the Spaniard's 10th in the competition for Liverpool.
Only a fine save from Rame denied Sissoko a first-ever Liverpool goal in the closing moments, before Hyypia somehow headed over a rebound off the goalkeeper from Gerrard's blast and substitute Robbie Fowler nodded wide from fellow sub Jermaine Pennant's clipped cross.
MAN OF THE MATCH
Luis Garcia
Two more goals to follow his strike against Aston Villa at the weekend
LIVERPOOL: Reina, Finnan, Carragher, Hyypia, Riise, Gerrard, Alonso (Zenden 57), Sissoko, Luis Garcia (Fowler 78), Kuyt, Crouch (Pennant 72). Subs: Dudek, Agger, Aurelio, Warnock. BOOKING: Sissoko.
BORDEAUX: Rame, Faubert, Jemmali, Cid, Marange, Menegazzo, Ducasse, Micoud (Mavuba 74), Wendell, Chamakh (Perea 11), Darcheville (Obertan 59). Subs: Valverde, Henrique, Alonso, Francia. SENT-OFF: Menegazzo (violent conduct).
REFEREE: Markus Merk (Germany).
ATT: 41,978.
NEXT GAME: Liverpool v Reading, Premiership, Saturday November 4, KO 3pm
WHAT DO YOU THINK? Have your say on our LFC forum - click here
