The defending League Cup champions advanced to the next round with victory at the Hawthorns, as the Turkish midfielder grabbed the headlines with two superb goals
The first half began with an error from both goalkeepers, as Brad Jones spilled a cross and allowed West Brom to take the lead through Gabriel Tamas before Ben Foster let a speculative Sahin shot squirm past him for the equalizer.
The second period was similarly frantic, and it took until the 82nd minute for the deadlock to be broken, as Sahin swept home his second to give the Reds the victory.
The game started badly for the away side, and it was behind in just the third minute. A free kick came into the Liverpool box, and was spilled by goalkeeper Brad Jones directly into the path of Gabriel Tamas, who drove the ball home comfortably.
It could have been worse for the Reds just a couple of minutes later, as Romelu Lukaku wrestled his way through their defense, but his shot was deflected behind by Jamie Carragher.
Liverpool was beginning to play with more confidence, and was rewarded for its enterprise in the 15th minute by another goalkeeping error, as Ben Foster allowed a speculative long-range shot from Sahin to squeeze past him for the equalizer.
The second half began frantically, with both sides having glorious chances as Stewart Downing missed a free header from a cross and Lukaku again wreaked havoc in the opening moments.
The game then descended into a similar pattern to that of the first half, with Liverpool enjoying more of the possession but with neither side able to create any clear chances, and struggling to move the ball beyond a congested midfield.
The Reds then had perhaps their best chance of the half in the 73rd minute, as Sebastian Coates flicked a dangerous free kick towards the goal, and was denied only by a reflex stop by Foster in the Baggies goal.
Liverpool then made history with the 80th-minute introduction of youngster Jerome Sinclair, who became its youngest-ever player in the first team at the age of 16 years and six days.
Just two minutes later, the Reds were celebrating again, as after some superb passing and movement, Sahin was on hand to sweep the ball home and give the away side a 2-1 lead, which it held on to for the final few minutes to advance to the next round.