What was ultimately a comfortable passage to the fifth round for the hosts might not have been but for three big moments either side of half-time.
Instead, a blistering finish by Dominik Szoboszlai and a charitable penalty for Mo Salah added to a Curtis Jones opener.
Jones turned home a cross from Milos Kerkez on 42 minutes to reward a home side who had got on top after a bright Albion start.
But the Seagulls blew two golden opportunities to respond right on half-time.
Jack Hinshelwood headed over from close range at a corner.
Then Diego Gomez shot weakly at Alisson after a Liverpool slip at the back.
The third big moment, soon after the break, was more about the excellence of Alisson as he pawed away a Lewis Dunk header which was destined to be turned home by Jan Paul van Hecke.
If hopes among the away fans were raised by that, they were deflated soon after when Szoboszlai was not tracked and thrashed home Salah’s lay-off on 56 minutes.
The third goal 12 minutes later almost felt like we were at a Salah testimonial.
Having shoved Ferdi Kadioglu out of the way, he went down when he felt minor contact from Pascal Gross and was handed the ball to send home the penalty awarded by Stuart Attwell.
Albion kept trying to find a way back into the game. Kadioglu saw an effort deflect over and Mats Wieffer made his return from injury in the closing stages.
But their fans, who kept up their support until the end, were left with nothing other than endeavour to applaud.