Lucas Leiva grabbed Liverpool’s winner as the Anfield side saw off the challenge of League One side Crewe.
Daniel Agger put the Reds ahead when he drove in a free-kick awarded for Michael O’Connor’s foul on David Ngog.
O’Connor atoned for his foul as he shot home from close range after keeper Diego Cavalieri parried Calvin Zola’s header from a Billy Jones corner.
But Leiva timed a run to perfection to meet a Jermaine Pennant cross and headed in to wrap up a Reds win.