It all meant it took around half-an-hour to open the scoring, Maxime Lucu taking a simple penalty chance following a Saracens offside, before another TMO decision prevented a Bordeaux try from the kick off.
The visitors lost lock Maro Itoje for 10 minutes following a deliberate knock on and that allowed Bordeaux to register their first counting try of the game, Buros making the break out wide and outside half Garcia picking up and finishing off.
Goode had the chance to whittle down that lead with a penalty at the start of the second half, but he hit the outside of the post from 40 yards.
Instead the breaks suddenly went the way of Bordeaux, number eight Tevita Tatafu managing to keep the ball alive before going to touch, the ball ricocheting off the knee of defender Ben Earl and being touched down over the line by Garcia.
Bordeaux snuffed out any remaining hope for the visitors as scrum power put Saracens on the back foot, allowing Garcia and right wing Damian Penaud to combine to put centre Nicolas Depoortere under the posts.
Bielle-Biarrey soon added a fourth from a loose ball bouncing off the head of centre Yoram Moefana, Lucu maintaining his perfect kicking record.
With the result in the bag Depoortere added his second from a slick tap penalty move, before Lewington grabbed a consolation try for the English team.
Bielle-Biarrey added his second to rub salt in the Saracens wounds, Tom Willis responding for the away side but it was too little too late.
