The English international sealed the precious win in the 58th minute to help Liverpool grab their first away win in the UCL this season.
The victory put Liverpool on top of Group C with seven points while Bordeaux remain fourth on just one point.
Bordeaux had the first real chance in the 25th minute minute when Johan Micoud failed to connect with Silva Wendell's free-kick as the ball eluded him with a few inches.
Crouch responded with a 36th-minute effort after he was picked out unmarked by Craig Bellamy inside the area but the giant striker failed to deliver as his first-timer went wide of the post.
But he made up in the 58th minute when he towered to head home Bellamy's corner-kick, sending the keeper the wrong way.
Elsewhere, PSV Eindhoven came from behind to earn a 2-1 away win over Turkish giants Galatasaray.
Sasa Ilic put Galatasaray in front in the 19th minute but Dutch Jan Kromkamp leveled the score in the 59th minute.
Ivory Coast international Arouna Kone sealed the victory for Eindhoven in the 72nd minute.
Eindhoven are second in Group C with seven points, behind Liverpool on goal difference.