The two teams began rather cautiously with few clear chances created in the first half hour aside with Brighton's Ferdi Kadioglu forcing Alisson Becker to make an early save and Curtis Jones narrowly missing the target with a curler.
The Reds thought they had taken the lead in the 36th minute but the referee ruled out Cody Gakpo's header from a Dominik Szoboszlai free kick due to an offside.
However, the home side eventually went ahead through the impressive Jones in the 42nd minute after diverting a Milos Kerkez delivery goalwards which went in off the bar.
The Seagulls almost equalized immediately but Jack Hinshelwood headed over the bar from a corner before Alisson denied Diego Gomez from close range to preserve the first half lead.
The Brazilian shot-stopper was called into action at the start of the second half, clawing away a goalbound Lewis Dunk header with Jan Paul van Hecke waiting for the tap-in.
In-form midfielder Szoboszlai doubled the lead for the Reds in the 56th minute with a drilled low finish following a Mohammed Salah through ball.
Twelve minutes later, Salah then added a third goal from the penalty spot after Kadioglu had fouled the Egypt star forward.
The visitors did apply more pressure in search of a way back into the tie, with Kaoru Mitoma and Georginio Rutter coming close as Liverpool held on to the clean sheet to progress to the fifth-round.
"We've improved," said goal scorer Curtis Jones after the match.
"We've had lads that have joined, we've had lads that have left. It has obviously taken time, but I think the team has gelled and the important thing is that off the pitch there is a good bond there as well."
The draw for the FA Cup fifth round is set to be conducted on Monday.