A strong Liverpool side had to wait until the second half to break down a stern Cardiff City side.
The Bluebirds, who sit 20th place in the Championship, made it very difficult in the first half as Liverpool missed a couple of sitters.
It did not take long though for the deadlock to be opened in the second half as Diogo Jota scored a header from a free-kick only eight minutes after the break.
Takumi Minamino got the second on his 50th appearance for the club whilst Harvey Elliott on his return from action scored his first goal for the club.
Cardiff City did get a consolation goal to make it 3-1 thanks to a powerful finish from Rubin Colwill.
Liverpool will host Norwich City in the 5th round of the FA Cup.
Caoimhin Kelleher: 5
Got away with a poor challenge in the second half which could have seen red due to his studs being high. Could be argued he should have done better for the Cardiff goal as well.
Trent Alexander-Arnold: 6
Was abject in the first half. Got the assist for the first goal with a world class delivery from a free kick. Had no problems defensively.
Ibrahim Konate: 5
Nearly gave away a penalty with some sloppy play. Could not get close to Colwill when he strook past Kelleher. Not his best 90 minutes.
Virgil Van Dijk: 8
Did not have too many duels to win but won all of them. Passing was a treat to watch again.
Kostas Tsimikas: 7.5
Great crosses all afternoon unlucky not to get an assist. Been superb all season.
Jordan Henderson: 6
Kept the midfield ticking over all afternoon in the regista role. Nearly got an assist with a lovely cross to Firmino.
Naby Keita: 5
Set up Jota for the first chance of the game and was unfortunate not to get an assist. Other than that, was not his greatest game however it was his first appearance since returning from AFCON.
Curtis Jones: 5
Missed a sitter in the first half after some nice build-up play from Minamino and Jota. Had one too many long shots which went well over the bar.
Takumi Minamino: 6.5
Struggled a little in the first phase of the game but grew into it. Was good in the half-spaces with intelligent play and emphatically scored the second goal.
Roberto Firmino: 5
Had a couple of headed opportunities which he could not make the most of. Also lost the ball in the lead up to the Cardiff goal. Some lovely skills on show yet again and was linking up with Luis Diaz well when he dropped into midfield.
Diogo Jota: 8
Been one of Liverpool’s best players this season and showed why yet again. Opened the scoring with a delicately placed header into the corner. On another day he would have had a hat-trick. His first chance was made from sheer strength and a ludicrous nutmeg, but he could not finish. Then after his first goal he had an acrobatic effort go just wide from a corner.
Harvey Elliott (Naby Keita 58): 7
Scored his first ever competitive goal for Liverpool on his return. A belter volley into the bottom right-hand corner. An emotional moment for the youngster.
Luis Diaz (Curtis Jones 58): 6
The first ever Colombian to play for Liverpool and he got an assist with some magical quick feet. A solid debut to say the least.
James Milner (Takumi Minamino 69): 5
Does what he does best and helped get the result over the line.
Andy Robertson (Kostas Tsimikas 69): 6
Got an assist with a nice delivery onto Elliott’s foot.
Thiago Alcantara (Jordan Henderson 78): n/a
Had no impact on the game which was already won. Great to see him back from injury.