Crouch struck in the 58th minute, heading home a Craig Bellamy corner, to keep Liverpool top of Group C with seven points and firmly on track for the knockout round.
Second-placed PSV Eindhoven also have seven points after their 2-1 win at Galatasaray, while the Turkish club and Bordeaux have a point each.
Liverpool, who travel to Old Trafford to face Manchester United in the Premier League at the weekend, were without talismanic captain Steven Gerrard.
