Liverpool debutant Samed Yesil wasted a headed opportunity before the visitors got back on level terms through Real Madrid loanee Sahin.
The Turkish midfielder collected the ball right of centre and drove at the West Brom defence before lashing a shot that managed to squeeze under Ben Foster and in.
The sides went in level at the interval and Liverpool almost completed the turnaround soon into the second half but Yesil's effort was blocked before bouncing just over the top.
Reds centre-back Sebastian Coates then saw an effort saved by Ben Foster before Sahin converted at the back post to win the game in the 82nd minute.
The draw for the fourth round takes place shortly.