The winger scored after being introduced as a first-half substitute as Jurgen Klopp's men delivered an improved display but lost Philippe Coutinho and Dejan Lovren to injury
Liverpool will head into the second leg of their League Cup semi-final tie against Stoke City with a slim advantage thanks to Jordon Ibe's first-half goal at the Britannia Stadium.
Ibe's cool finish after Joe Allen had diverted Adam Lallana's cut-back into his path proved decisive for the eight-time winners in a match of few clear-cut opportunities.
On a run of just two wins in seven games, Liverpool produced a much-improved performance to head back to Merseyside with a lead ahead of the return fixture on January 26.
The only negative for Jurgen Klopp was the lengthening of an already problematic injury list, with Philippe Coutinho and Dejan Lovren both substituted due to hamstring complaints.
Stoke, who are bidding to make the final for the first time since 1972, had more possession but struggled to make an impact going forward, with substitute Jonathan Walters wasting a glorious late chance.
Mark Hughes's men were unchanged while Klopp made three alterations, bringing in Kolo Toure, Allen and Lallana as Christian Benteke dropped to the bench and Mamadou Sakho missed out through injury.
Liverpool were dominant in the early stages, with Roberto Firmino and Lallana keeping Jack Butland busy with two saves from powerful long-range efforts.
The visitors' momentum threatened to stall when Coutinho limped off after attempting a cross and Lovren pulled up soon after following a sliding tackle.
But Liverpool got the goal their play deserved in the 37th minute. James Milner, on for Lovren, released Lallana down the right and his cut back to Allen was sliced across the six-yard box to Ibe, who made no mistake from close range.
Stoke's best chance of the half went begging when Marko Arnautovic headed wide from close range after Glen Johnson's cross from the right.
Just before the break, Simon Mignolet had to get down well to deny Johnson as he stabbed a shot through a crowded box.
Referee Anthony Taylor was unmoved when Glenn Whelan went down under pressure from Alberto Moreno early in the second half as the hosts began to look more threatening.
At the other end, Allen was denied a spot-kick after being bundled over by Whelan as Milner saw a goal-bound shot blocked on the line by Erik Pieters.
Stoke pressed again when Joselu's shot deflected off Toure and just over the crossbar, but the best chance was still to come.
The injury-hit visitors had Toure limping through the final few minutes with cramp and almost paid the price when Walters raced through to shoot wide of the far post at the death.