In a game with little goalmouth quality, a spectacular Harvey Elliott effort in the first half saw Jurgen Klopp's rotated side avoid a second successive defeat.
The Reds have been rewarded for their efforts with another daunting trip to Brighton & Hove Albion, who trounced Klopp's men 3-0 in the Premier League last weekend.
That was one of five FA Cup ties to take place on Tuesday evening, with Birmingham City prevailing in their rescheduled match with Forest Green Rovers 2-1 at the New Lawn.
The League One side got off to the perfect start in the third-round tie, which was originally postponed last weekend due to a waterlogged pitch, as Ben Stevenson struck the opener with eight minutes gone.
However, second-half goals from Lukas Jutkiewicz and Kevin Long turned the game on its head, as Birmingham booked a trip to fellow Championship side Blackburn Rovers in round four.
There was last-minute drama at the DW Stadium in Wigan Athletic's clash with Luton Town, as the Hatters struck in the final few seconds to clinch a 2-1 win after coming back from a goal down.
Wigan drew first blood through Thelo Aasgaard early in the second half, but Cauley Woodrow levelled not long after before Elijah Adebayo's winner in the eighth minute of second-half injury time.
Luton have been rewarded with a home tie against Grimsby Town in the fourth round, but non-league Chesterfield's run came to an end in comprehensive fashion with a 4-0 loss to West Bromwich Albion.
John Swift opened the floodgates in the first half before Tom Rogic, Jake Livermore and Jovan Malcolm got in on the act, as the Baggies prepare to meet Bristol City for a place in the fifth round.
The Robins' replay with Swansea City went the distance, as Mark Sykes put Bristol City in the ascendancy before Oliver Cooper's response on the 73-minute mark forced extra time.
However, twenty-year-old Sam Bell - who had set up Sykes's opener - turned goalscorer in the 112th minute to confirm Bristol City's place in round four.
Meanwhile, Accrington Stanley's replay with Boreham Wood was postponed due to an unplayable pitch, and the two teams will instead renew hostilities on Tuesday January 24 for the chance to meet Leeds United or Cardiff City in round four.