The first 24 hours of sales saw 2,000 tickets snapped up and that figure has continued to grow, encouraged by a subsidy on prices.
Albion decided to under-write the agreed cost of tickets for season ticket holders and existing 1901 Club members.
That means adults pay £20 instead of up to £34, with age-group concessions also available.
About 1,000 Seagulls fans cheered their side to a 2-1 win at Manchester United in the third round, with the number trimmed by reasons including cost and scheduling.
That number has already been more than doubled for the fourth-round tie, which has been scheduled for the unappealing 8pm slot on a Saturday night.
Albion will be playing on Valentine’s Night for the second season in succession, a year on from their 3-0 win over Chelsea at the Amex.
Fans who buy a ticket for the Liverpool tie will receive a unique code for a £20 discount on any sized home shirt, subject to availability.
The club expect to revert to their traditional blue and white having worn a purple change strip on their previous trip to Anfield this season.
Jenny Gower, Albion’s head of ticketing and supporter services, said: “We’re delighted with the response from our supporters, and that the subsidised offer has proved so popular.
“We really appreciate the efforts of our fans as we know how challenging a trip this is in terms of their time and financial commitment.”