Sagbo made the most of a rare start in the absence of the cup-tied Shane Long and Nikica Jelavic, slotting an 85th-minute equaliser to cancel out Leonardo Ulloa's first-half goal at the Amex Stadium.
Brighton were on track for a meeting with former manager Gus Poyet – now in charge of Sunderland – in the sixth round but paid the price for sitting back in the latter stages.
Sagbo and Curtis Davies were denied by the woodwork for Hull in the first half but Steve Bruce's side eventually got their reward.
And they could have won it in stoppage time, only for David Meyler's header to be cleared off the line by Adam Chicksen.
Poyet was controversially sacked for gross misconduct by Brighton in July after the club's failed campaign in the Championship play-offs.
Brighton controlled the early proceedings but chances at both ends were limited until Sagbo's acrobatic volley crashed into the crossbar in the 27th minute, with goalkeeper Peter Brezovan well beaten.
The Argentinian's clever backheel played in Buckley, and he returned the favour with a well-weighted pass that sent Ulloa through on goal.
Hull could have drawn level within two minutes only to see the woodwork again come to Brighton's rescue.
This time Davies powerfully met a Huddlestone corner but saw his effort thud into the crossbar, while Sagbo's headed follow-up was nearly diverted in by Matthew Upson, who saw his attempted clearance fall into the arms of Brezovan.
Ulloa really should have doubled Brighton's lead on the stroke of half-time, but inexplicably nodded wide when free at the far post.
Hull were slow to get going in the second half and even the introduction of Matt Fryatt - who scored twice at Southend United in the fourth round – and George Boyd failed to rouse them.
