The Seagulls looked set to reach the last-eight juncture for the first time since 1986 thanks to Leonardo Ulloa's quality first-half goal, only to see a mouth-watering home clash with Sunderland and former manager Gus Poyet put on hold.
The Uruguayan's three-and-a-half year spell with the Seagulls came to an acrimonious end in the summer, with his first return to the Amex Stadium dependent on whether Brighton can win the replay after Sagbo's late strike made it 1-1.
It led to a frantic end to what had been for the most part a disjointed match, thanks in no small part to the sides' combined 13 changes for the fifth-round clash.
There were few chances of note until Sagbo stretched to send an unconventional volley onto the crossbar in the 27th minute - a strike that appeared to kick-start proceedings.
Just three minutes later Brighton went ahead through Ulloa, who played an impressive one-two with Will Buckley before dinking over Allan McGregor.
Hull threatened an immediate leveller as Curtis Davies' header hit the woodwork, but that was as close as Steve Bruce's side came until five minutes from time when Sagbo struck home after collecting a cross from Sone Aluko, making his first start after four months out with an Achilles injury.
Both teams had late chances to snatch victory, but Davies and Jake Forster-Caskey were both in the right place to clear headers off the line.
Given the number of changes made by both teams, it was little surprise that it took a while for the match to get going.
Albion edged the opening 20 minutes but were struggling to create clear-cut chances, with Forster-Caskey hitting wayward efforts either side of an equally off-target Buckley strike.
A tame Sagbo attempt all Hull could muster until the 27th minute, when the powerful striker met a cross from the right with an unorthodox volley that rattled the underside of the crossbar.
The effort started an exciting period in which Brighton broke forward to net a stylish opener.
Having started the move with a neat flick to Buckley, forward Ulloa collected the return ball and burst forward on goal, continuing to delicately lift over the onrushing McGregor.
It was a wonderful team goal by the hosts and one that was so nearly chalked off immediately as Tom Huddlestone's corner was directed onto the crossbar by Davies. Sagbo's follow-up looked set to be turned into his own goal by Matthew Upson, only for goalkeeper Peter Brezovan to save his blushes.
Play was delayed after Adam Chicksen collided with Davies, before Ulloa had the home fans on their feet in stoppage time.
Unmarked, he met a fine right-wing cross with a powerful header that rippled the net - although it quickly became apparent that it had only done so after going narrowly wide and rebounding off the stanchion.
Solly March replaced Buckley when the teams returned for the second half, with the midfielder welcomed to the fray by a challenge that earned Hull captain Robert Koren a yellow card.
City were struggling to create much on the attack and brought on George Boyd and Matt Fryatt shortly after Keith Andrews turned behind a fizzing Huddlestone free-kick.
The changes did not have the desired effect, though, with Inigo Calderon having efforts either of a poor March strike as Brighton looked to double their advantage.
Hull were becoming desperate in their bid to level, with Fryatt and Huddlestone leading the visitors charge without success.
The 695 travelling fans continued to gee on their side as the match entered the closing stages and got their reward with five minutes left.
Aluko whipped in a hopeful cross from the right, which Sagbo controlled under pressure and slotted through Brezovan's legs.
Both sides had wonderful chances to win after that, with Brighton seeing a goalbound header cleared off the line by Davies before City came close at the other end.
Koren saw a late short deflected for Bruce's side, before Forster-Caskey cleared substitute David Meyler's header off the line.