Luis Garcia scored twice and Steven Gerrard netted his first goal of the season as Liverpool qualified for the Champions League knockout stages.
Garcia netted a goal in each half to help Liverpool to victory
Garcia volleyed the Reds in front on 23 minutes with a powerful shot from a tight angle after Gerrard's cross.
Bordeaux's Fernando Menegazzo was sent off for a savage headbutt on John Arne Riise on 67 minutes, before Gerrard sidefooted home from inside the box.
And Garcia raced clear on 76 minutes to shoot home and seal the victory.
It was a deserved win for Liverpool, who overcame a shaky spell early in the second half to guarantee progress from Group C.
The Reds did look rather disjointed in the opening stages - somewhat ironically given that boss Rafael Benitez named an unchanged side for the first time in 100 games.
But Garcia gave the home fans something to cheer with their first shot on target.
The Spaniard produced a powerful volley from a tight angle on the left, hitting Gerrard's cross into the ground and over Bordeaux keeper Ulrich Rame.
The goal seemed to settle Liverpool, who began stroking the ball around far more fluently.
But first-half chances were at a premium with Liverpool looking comfortable against a toothless Bordeaux attack.
Wendell's free-kick just before the break - comfortably saved by Jose Reina - was as close as the visitors came.
But Bordeaux showed far more attacking intent after the interval, Florian Marange setting the tone with a 47th-minute drive which Reina had to tip over the bar.
Jean Claude Darcheville also had the alarm bells ringing in the home defence as he beat the offside trap, only to be denied by Reina.
Johan Micoud was also denied by a fine lunging tackle from Sami Hyypia as Bordeaux enjoyed a spell of pressure.
Liverpool were struggling to rediscover their form, but midway through the half Menegazzo handed the initiative back to the Reds after his baffling act of thuggery left his side a man short.
Bordeaux had been unhappy with a Mohamed Sissoko challenge in midfield and during the break in play, Menegazzo strode up to Riise and headbutted the Norwegian.
Riise was bleeding profusely and referee Markus Merk swiftly produced a red card.
Shortly afterwards, Gerrard found the net to double the lead, finishing smartly after Boudewijn Zenden's superb pass.
And Garcia strode through on 76 minutes after a defensive blunder before rifling home to end the game as a contest.
Liverpool created more chances late on, as Sissoko's drive was tipped over by Rame, while Hyypia headed over from the rebound after the keeper parried Gerrard's shot.
Liverpool: Reina, Finnan, Carragher, Hyypia, Riise, Gerrard, Alonso (Zenden 57), Sissoko, Luis Garcia (Fowler 78), Kuyt, Crouch (Pennant 72).
Subs Not Used: Dudek, Agger, Aurelio, Warnock.
Goals: Luis Garcia 23, Gerrard 71, Luis Garcia 76.
Bordeaux: Rame, Faubert, Jemmali, Cid, Marange, Menegazzo, Ducasse, Micoud (Mavuba 74), Wendell, Chamakh (Perea 11), Darcheville (Obertan 59).
Subs Not Used: Valverde, Henrique, Alonso, Francia.
Sent Off: Menegazzo (67).
Ref: Markus Merk (Germany).