Albion went ahead through Gabriel Tamas but on loan Real Madrid midfielder Sahin levelled before half-time and struck the winner with eight minutes remaining.
Liverpool's first choice line-up had been humbled 3-0 at Albion on the opening day of the season in the Premier League but their potential stars of the future did them proud on Wednesday night.
Rodgers made 11 changes from the side beaten at home by Manchester United on Sunday but his side were worthy winners after an uncertain start.
Oussama Assaidi caused all sorts of problems down the left flank for Billy Jones and then Craig Dawson and set up Sahin's late decider.
Turkish international Sahin also showed his quality in the centre of the park.
The home side will be bitterly disappointed at their exit after fielding a strong side but only midfielder Youssouf Mulumbu and striker Romelu Lukaku emerged with much credit.
Albion needed only three minutes to take the lead after a glaring error by Reds goalkeeper Brad Jones.
Liam Ridgewell floated a harmless looking free-kick into the Liverpool box from the left flank but Jones, under no pressure, dropped the ball and central defender Tamas kept his volley down as it flew into an unguarded net.
West Brom came close to doubling their lead when Lukaku cut in from the left but his shot was deflected wide by Jamie Carragher.
Albion dominated the early exchanges and a fierce drive from Markus Rosenberg forced a low save out of Jones after Lukaku had shaken off the challenge of Carragher.
But Liverpool should have levelled after 11 minutes when debutant Samed Yesil headed wide from Assaidi's centre.
Another goalkeeping error, this time by Ben Foster, led to Liverpool equalising after 17 minutes through Sahin
Sahin let fly with a speculative low 30-yard drive and Foster allowed the ball to slip under his body and into the net.
Ridgewell limped out of the action after 22 minutes and was replaced by Craig Dawson
Liverpool were now in the ascendancy with winger Assaidi causing constant problems but Albion still posed a threat on the counter-attack.
Jones raced out of his goal to fly hack clear from Lukaku, who then tested the Liverpool keeper with a low drive.
Mulumbu and Reds defender Andre Wisdom were booked after challenges on Sahin and Billy Jones respectively.
Liverpool dominated the early exchanges of the second period and Yesil's low shot hit the leg of Foster and somehow bounced up over the bar.
Then the former England keeper could only stand and watch as Jordan Henderson's curling attempt from the left of the Albion box struck the bar.
Rosenberg was struggling to make an impact but did force Jones to turn aside an effort from on the edge of the Liverpool area.
Foster again struggled in dealing with a dipping drive from Daniel Pacheco and was having one of his less than convincing games - but he partially redeemed himself in denying Sebastian Coates at close range.
Jerome Sinclair became Liverpool's youngest ever player at 16 years and six days when he was introduced as an 81st minute substitute, beating the record of Jack Robinson (16 years and 250 days).
Then a minute later Sahin won it for the visitors when he found space at the far post to convert Assaidi's low cross.
|West Bromwich Albion||Team Statistics||Liverpool|
|1||1st Half Goals||1|
|5||Shots on Target||6|
|2||Shots off Target||5|