Yves Bissouma gave the Seagulls a 28th minute lead, blasting a shot from 30 yards out before Gary Madine levelled in first half stoppage time.
The match was decided in the 58th minute when Steven Alzate scored his first goal in a Brighton shirt after seeing an Alexis Mac Allister shot take a deflection off of his back.
Story of the match
Blackpool were severely depleted as five members of the first team tested positive for COVID-19 and in the third minute they were almost caught out. Mac Allister's corner found Lewis Dunk, who headed onto the roof of the net.
Again, the Argentine looked to be the provider as on 15 minutes he delivered another excellent corner only to see an unmarked Adam Webster head over.
Three minutes later, the visitors came close to taking a shock lead. Luke Garbutt struck a free kick from 25 yards out, drawing a save from goalkeeper Christian Walton, who pushed it wide.
Brighton were then in front as Bissouma powered home an unstoppable effort past Tangerines goalkeeper Chris Maxwell and into the top left corner from 30 yards out.
The hosts had a chance to extend their advantage as Zeqiri fired wide from Joel Veltman's cross and he also fired wide from the left corner of the box.
There was little in the way of a threat posed by Blackpool, but they were on equal terms as Jerry Yates played a cheeky back-heel to Garbutt, who produced a first-time cross that Madine beat Dunk to, steering past Walton for his fourth goal of the competition.
He almost doubled his tally in the 53rd minute when he couldn't make contact with James Husband's cross. Mac Allister then forced Maxwell into a diving save at the other end.
Moments later, the Argentine had another go from the edge of the box and the shot took a deflection off of Alzate's back, the ball going one way, Maxwell diving the other to give Brighton a 2-1 lead.
Substitute Neal Maupay had a chance to put the game to bed for the Seagulls as he tested Maxwell from a Leandro Trossard cross, but the keeper was up to the task. Pascal Gross then fired over and Solly March curled one wide.
Maxwell then denied Davy Propper in the last action of the game as Brighton advance to a fifth-round clash against either Brentford or fellow Premier League side Leicester City.
Man of the match: Alexis Mac Allister
He was involved in all of the key moments for Brighton as he tested the Blackpool goal early on, provided some excellent service with his crosses and started the move, even by accident, that wound up deciding the match.