Rafa Benitez made 11 changes to the side that was frustrated in a goalless draw against Stoke on Saturday.
The unfamiliar team of youngsters and expensive outcasts could have come close to a similar surprise after being held level for part of the game.
Defender Daniel Agger opened the scoring for the home team on 15 minutes with a driven free-kick from 25 yards out.
Ten minutes later, Crewe pushed forward for the first time and Diego Cavalieri, who was making his Liverpool debut between the sticks, fumbled a ball behind that was seemingly running out for a goal-kick.
The keeper almost made amends from the resultant corner as he pulled off a fine save from a Calvin Zola header but couldn't keep hold of the ball.
Influential Crewe midfielder Michael O'Connor was waiting in the wings of the rebound to poke home an equaliser for the visitors.
Up until this point this unaccustomed Liverpool side looked at relative ease.
Jermaine Pennant took advantage of his experience and comfortably roamed the Railwaymen's half throughout the opening period.
Summer signing Philipp Degen, who was making his first start, excited the home supporters on more than one occasion with his willingness to push forward from his full-back position.
The Premier League team went in level at the break, but it was just before the hour-mark that the Reds regained the lead.
A Pennant cross found Lucas and the Brazilian midfielder's looping header somehow found its way in between Crewe keeper Steve Collis and the crossbar.
Not resting on his laurels after an earlier scare, Benitez introduced Fernando Torres on 67 minutes replacing David N'Gog as Liverpool looked to finish off the tie.
It was the Spaniard who came closest to increasing their advantage with a pair of chances in the final ten minutes of the match.
He first forced Collis into a low save from a header, then moments later he found the keeper an obstacle once again with a one-on-one that he'd normally finish with ease.
Three minutes from the whistle, Crewe should have equalised for a second time.
Forward Tom Pope spurned a great chance from only yards out when it surely would have been easier to find the back of the net.