After an end-to-end contest that saw the hosts miss a plethora of big chances, it was a Jakub Kiwior own goal that broke the deadlock for the 10 minutes from time, diverting a whipped Trent Alexander-Arnold free-kick into his own net to give the visiting Reds a precious lead.
The win confirms Liverpool's place in the fourth round of the FA Cup, with the result crucially avoiding the addition of an unwelcome replay to the Reds' already busy calendar.
Now, Liverpool wait to hear who they'll face in their cup campaign - something they will find out in the fourth round draw, which takes place at 7.50pm GMT on Monday.
The draw will air on free-to-view TV in the UK, being shown on ITV1 ahead of Manchester United's tie with Wigan Athletic.
The likes of Manchester City, Chelsea and Tottenham will be in the draw, as well as lowest-ranked side Maidstone United - the National League South outfit that grabbed headlines this weekend after toppling League One play-off hopefuls Stevenage to progress.
Fourth round fixtures are set to be played across the final weekend of January.