It was an ugly win, but a win nevertheless and Brendan Rodgers will be happy to have bounced back from last Saturday's 2-2 draw with Aston Villa with a win and a clean sheet.
Here, we talk through six Liverpool talking points from the match, which produced a mixture of positives and points for progression as the Reds look to Tuesday's Merseyside derby against Everton.
No sooner had the game got underway than Aly Cissokho gave the ball away from a throw-in on the half-way line.
The Frenchman looked lost and confused, consistently losing possession and rarely sparking any confidence in Liverpool's left-sided attacks.
Rodgers' comments on Thursday, as per This Is Anfield, through gritted teeth on how Cissokho is "performing to the best of his capability" were a telling sign that the Reds' boss can't wait for Jose Enrique's return from injury next month.
A much-improved performance from Victor Moses may convince some Liverpool fans that he still has a chance to succeed at the club and potentially make his loan move from Chelsea permanent.
A tricky run down the left, cutting in and crossing low towards Daniel Sturridge almost resulted in a goal early on for the Reds, and he was arguably Liverpool's best player in the first half.
The Nigerian's snap shot from the edge of the area on 26 minutes to give Liverpool the lead was the type of out-of-the-blue goal Liverpool need more often, coming against the run of play after an impressive opening from the Cherries.
The goal gave the 23-year-old a spark of confidence, providing better movement and spark in Liverpool's attacks. It was a far cry from the aimless figure who wandered the pitch against Oldham in the previous round.
After a very disappointing first half of the season for the Reds, Moses now has the opportunity to build on Saturday's game and prove his critics wrong.
Brad Jones made a rare start between the sticks for Liverpool, with Simon Mignolet given a rest after some uncharacteristic mistakes in Premier League games lately.
However, on several occasions, the Australian keeper looked cagey coming out of his goal to collect balls.
During the first half, Jones flapped at a low cross, eventually getting a fist to the ball to divert it away from the danger zone. However, Jones was under no pressure and a better keeper would have collected the ball safely and with ease.
Whilst Liverpool are fortunate enough to have the reliable Mignolet week in, week out, should injury force the Belgian out of action, Jones' name on the team sheet every week may not strike much confidence into Liverpool's back line.
Whilst Philippe Coutinho offered some brief flashes of brilliance with his close controlled runs from midfield towards the attack, his end product is still lacking.
In comparison to Raheem Sterling, he lacks the confidence to take players on or the conviction to play the dynamic through balls he was producing last season. The Brazilian has been under-performing for some time for Liverpool.
That's not to say that Coutinho had a bad game, but some time out of the starting line-up to reflect on his whirlwind year since joining from Inter Milan would be healthy right now. It's easy to forget he's still only 21 years old.
In the crunch Premier League clash at home to Everton in midweek, in-form Sterling will most likely start in a three-pronged attack with Luis Suarez and Sturridge rather than Coutinho.
Martin Kelly's performance at right-back was solid enough, but the 23-year-old is still struggling to complete 90 minutes.
With Glen Johnson out injured for an indefinite period of time, Liverpool are looking short of options on both sides of defence.
Jon Flanagan's second-half substitute appearance, his first action since picking up a hamstring injury before Christmas, comes as good timing ahead of the Merseyside derby.
The youngster looked sharp and in control during his brief appearance at the Goldsands Stadium and will fit nicely back into Liverpool's defence, most likely at right-back, on Tuesday night.
It may not have been the prettiest of victories for Liverpool, but it was never going to be a nice afternoon out at the seaside for Rodgers' men.
The Reds never really got out of third gear but improved as the game went on, and the home side's fatigue began to set in.
Liverpool went to the south coast on Saturday looking simply for a win and a place in the fifth round of the FA Cup and they got it.
Rodgers will unlikely lose much sleep over his side's overall performance at the Goldsands, and Martin Skrtel's second-half head injury shouldn't affect plans either.
Now attention turns to the more important matter of Everton at home in the Premier League in three days' time when Rodgers will welcome back Sterling and Mignolet to the starting line-up.