The Netherlands international, who scored twice in the Reds’ 5-1 third-round success over West Ham, took his career tally in the competition to eight goals in as many games with two strong finishes.
Simon Adingra pounced on the rebound 10 minutes from time to halve the deficit before Luis Diaz quickly restored the two-goal lead for Arne Slott’s side before Tarik Lamptey’s deflected consolation.
Reds goalkeeper Vitezslav Jarosz produced a fine save to turn Adingra’s header against the post at 1-0, while Lamptey and Evan Ferguson also wasted good openings for Albion.
After a forgettable first half, the Seagulls were the dominant force in the 17 minutes between Gakpo’s goals but were mercilessly punished as their bid to reach the last eight of the tournament for the first time in 46 years fell short.
Andy Robertson captained the Reds after being one of just three players retained from the squad that started Sunday’s 2-2 draw with Arsenal, while Brighton also made eight changes.
Seagulls goalkeeper Jason Steele, who captained the home side, survived some nervous early moments before Lamptey wasted a golden chance to open in the 13th minute.
After Jan Paul van Heke’s searching pass picked out Adingra, marauding right-back Lamptey got on goal only to fire straight at Liverpool’s Czech keeper Jarosz.
Gakpo then curled wide at the other end moments later as a crafty first half briefly threatened to burst into life.
Liverpool, second in the Premier League table, have lost just once since Dutchman Slott replaced Jurgen Klopp in the summer.
The 10-time League Cup winners were twice denied an opener before the break when Steele deflected Dominik Soboslay’s cross and denied Robertson from a tight angle.
Gakpo broke the deadlock in style just 30 seconds into the second half.
The former PSV Eindhoven striker was released down the left by Tyler Morton and in front of the traveling fans he cut through Jakub Moder to unleash a spectacular strike into the far corner from just inside the 18-yard box.
Brighton responded well to falling behind and were unfortunately unable to equalize straight away.
Full debutant Jarosz produced a superb save to tap Adingra’s header onto the right-hand post after Lamptey broke in from the right, before Ferguson blazed over the crossbar and Brian Gruda curled over.
Albion were on top but were caught out in defense as Liverpool doubled their advantage.
Two minutes after Steele saved from Szoboszlai to prevent the Reds from capitalizing on a three-on-one counter-attacking situation following a Lamptey error, Gakpo grabbed his second.
Lamptey was again at fault, passing possession deep into his own half, which allowed Gakpo to advance on goal and viciously chip Steele at the near post.
Brighton blew a two-goal lead late in Saturday’s 2-2 draw with Wolves and threatened to turn the tables by pulling one back.
After Jarrell Kwanza carelessly gave the ball away as Liverpool tried to play from the back, Ferguson’s shot was parried by Jaros, only for the alert Adingra to finish off the loose ball.
Hopes of a fightback looked quickly extinguished after Diaz fired low into the bottom left corner and over the bar in the 85th minute.
However, Lamptey increased the tension five minutes into stoppage time when his long-range shot flew wide of Cuanza before the Reds stayed on course to retain the trophy.