Rhys Williams’ clinical header opened the scoring during an evenly-matched first half at the Totally Wicked Stadium.
Paul Glatzel doubled the Reds’ lead just before the interval and the striker added two more in the second period to seal a hat-trick and rack up a comprehensive scoreline.
Barry Lewtas’ in-form team therefore progress to a third consecutive home match in the competition this season, against Wigan Athletic.
Accrington should have taken the lead within two minutes of kick-off.
Lewis Gilroy made ground at the left edge of the Liverpool area and his diagonal cross to the far post found Charlie Ridge unmarked yards from goal and primed to net – but he lifted his shot high over the bar.