كتب : Ahmad Saied
The former European champions looked on their way to a comfortable victory after going three up thanks to a Peter Crouch brace and Luis Garcia's header.
Galatasaray stormed back to the game with two powerful headers from substitute Umit Karan with about 25 minutes remaining on the clock.
But Liverpool held on to their slim lead to secure a nervy win and go top Group C with four points, leading PSV Eindhoven, who have the same tally, on goal difference.
Third-place Galatasaray and rooted Bordeaux are level with one point each.
Crouch celebrated his return to the starting line-up as early as the ninth minute when he converted a Fabio Aurelio cross home with a neat one-touch.
Garcia doubled the lead after just five minutes, powering a brilliant header into the back of the net after Jermaine Pennant's cross left the keeper stranded.
English international Crouch nailed Liverpool's third and the game's best goal seven minutes into the second period when he slammed the ball with an acrobatic shot after connecting with a perfectly-weighted cross from Steve Finnan.
But as the huge Anfield crowd was growing in confidence, Karan, who replaced Cihan Haspolatl in the beginning of second-half, ran riot to spark the Turkish side into life.
He headed Arda Turan's cross from the left flank past Jose Reina in the 59th minute before towering to another cross from the right to steer another header into the back of the net six minutes later.
In another game, Mika Vaeyrynen scored a debatable goal five minutes after the hour mark to give PSV Eindhoven a precious 1-0 victory over hosts Bordeaux.
Aruna Kone looked offside before he fed Vaeyrynen who knocked the ball home, but the assistant referee didn't flag the Ivory Coast striker.