The Reds fell behind early in the game when Gabriel Tamas fired the home side in front. Sahin pegged Steve Clarke's side back in the 17th minute with a long-range effort and then, with less than 10 minutes left on the clock, the Turkey international tapped home Liverpool's winner.
It had been a dream start for West Brom when Tamas arrowed the ball through a wall of Liverpool defenders after Brad Jones spilled a high cross into his path in the third minute.
Liam Ridgewell floated a pass into the heart of the Reds' penalty area and Jones rushed out to intercept; however, the 'keeper spilled the ball into the feet of Tamas, who shot through a wall of defenders and into the empty net behind.
Moments later Romelu Lukaku went close to doubling West Brom's advantage. The Belgian darted past Jamie Carragher but the veteran Liverpool defender made amends for his momentary lapse in concentration by hurling himself in front of the striker's attempted finish.
But Lukaku was wreaking havoc upon the Reds' defence, who appeared jarred by his sheer presence. He broke free once again in the seventh minute and pulled the ball back for Markus Rosenberg, who fired low, but Jones was down smartly to get his body behind the powerful effort.
The Reds had a fantastic opportunity to level the game when Oussama Assaidi flew down the wing, checked back before injecting pace and beating his man to the by-line. From there, he curled toward Ben Foster's near post but the arriving Yesil was unable to glance the ball home.
The move may not have pulled the Reds level but it galvanised Rodgers' side.
The away fans, packed into the stand behind Jones's goal, played their part as they chanted "One Brendan Rodgers" in a passionate show of support for the manager and, within a minute, Sahin had fired past Foster to level proceedings.
Stewart Downing found Andre Wisdom out on the right wing, and the young full-back showed composure when he rolled the ball to Sahin, who took a touch before smashing home from 30 yards.
A tempestuous minute of play, just prior to the interval, saw Wisdom flick a powerful free kick yards wide of his own goal before Jack Robinson hooked the ball to safety and the Reds broke. Assaidi fed Yesil, who was visibly caught as he attempted to break free of the West Brom defence, but the young striker remained on his feet before Jonas Olsson dispossessed him.
The Reds started sprightly after the interval. First Daniel Pacheco and Yesil exchanged passes on the edge of the West Brom box, but Yassine El Ghanassy poked the ball clear as the Spaniard shaped to shoot.
Then Assaidi crossed deep towards the back post, Billy Jones ducked out of the cross but the loose ball caught Downing by surprise as he arrived at the far post to nod home.
When Yesil spun and fired at Foster from range, the ball bobbled between the stopper's legs before bouncing over the crossbar.
Sahin was dominating proceedings in the middle of the field, and the Reds were beginning to play some expansive football.
Pacheco went close with two efforts. First he twirled past a marker after bringing the ball down with a delicate touch before looping a shot over Foster and onto the crossbar. Then he fired low and hard at the 'keeper, who managed to push his goal-bound effort wide.
Sahin whipped a tantalising pass in behind a back-tracking Wrest Brom defence and Sebastian Coates barged his way to the front of the pack to divert the ball towards goal, but Foster was there once more.
With 10 minutes of normal time remaining, 16-year-old Jerome Sinclair became the youngest-ever Liverpool player when he replaced Yesil, and Suso came on in place of Pacheco.
And within seconds Suso unlocked the West Brom back four sliding a pass into Assaidi, who squared play for Sahin to tap home.
Liverpoolfc.com journalists' man of the match: Nuri Sahin.
Photo via Liverpoolfc.com