An early red card to Mason Holgate was a harbinger of doom for the Blades, who conceded twice to Africa Cup of Nations winner Simon Adingra, while Danny Welbeck, Facundo Buonanotte and a Jack Robinson own goal also contributed to the rout.
As well as suffering a first-ever Premier League defeat to the Seagulls, Sheffield United also became the first side in the history of the English top flight to concede three goals in five successive home games in all tournaments, as their seemingly inevitable demotion back down to the Championship draws ever closer.
After returning from the Africa Cup of Nations as a continental champion, Adingra was immediately drafted back into the Brighton XI in place of Adam Lallana, while Will Osula led the line for Sheffield United due to the absence of Cameron Archer.
Despite coming into the game on the back of a surprise 3-1 success over Luton Town, not since 2020 had Chris Wilder's side won consecutive Premier League matches, and their hopes of ending that barren run took a major hit just 13 minutes in.
Stuart Attwell did not immediately brandish the red card for a horrendous Holgate challenge on Kaoru Mitoma, but upon a check of the monitor, the referee immediately sent off the ex-Everton defender, who caught Mitoma on the knee with his studs and could have no complaints.
Brighton were already making inroads when the contest was 11v11, and the Seagulls needed just seven more minutes to make their extra man count, as Lewis Dunk headed a corner into the path of Buonanotte, whose close-range attempt struck Osula on the line and rippled the roof of the net.
There was some initial confusion whether the goal had been given as Wes Foderingham quickly hooked the ball away, but Brighton could eventually revel in their opening goal, although their joy was very nearly short-lived as Gustavo Hamer's slick solo run ended with the midfielder firing wide from a tight angle in the 22nd minute.
A frenzied period then saw James McAtee escape giving away a penalty for a two-handed push on Mitoma in the 23rd minute, but just one moment later, Brighton fans had their arms raised in a more celebratory fashion.
A dipping Pascal Gross delivery to the back post landed at the feet of Mitoma, whose attempt was palmed back into the danger area by Foderingham, and Welbeck was well-placed to fire the ball into the roof of the goal.
Wilder eventually bit the bullet and sacrificed Osula for another defender in Auston Trusty, but there was a brief ray of hope for the hosts in the 43rd minute, as a penalty-box scramble from a corner ended with Ben Osborn hooking the ball in from close range.
However, Attwell's whistle blew immediately, and the referee's decision was upheld following a VAR review, although it was unclear whether a foul or offside offence was the reason for the decision to disallow the goal.
Brighton were certainly not home and hosed by the half-time whistle, and Roberto De Zerbi opted for a double change at the break, introducing Igor and Jakub Moder for Tariq Lamptey and Adam Webster.
Evan Ferguson also entered the fray in the 66th minute to try to inject some life into a relatively flat second half, and the Irishman posted a shot on target within seconds of coming on, but his header from a corner was straight down the throat of Foderingham.
Sheffield United remained camped inside their own defensive third as De Zerbi's side continued to probe, and while the Blades defended diligently with 10 men, the result was set in stone in the 75th minute.
A livewire all afternoon, Mitoma sent in an inswinging delivery from the left, and Jack Robinson stuck out a leg to try to divert the ball behind, but the Blades defender poked the ball into his own net to add to Sheffield United's defensive horror show.
The home fans had already begun to make a beeline for the exits by the time Brighton struck their fourth in the 78th minute, as Gross sent in a low ball from the right for Adingra, who took it first time on the volley and found the far corner with a well-executed side-footed strike.
Another seven minutes passed before the AFCON winner doubled his tally for the afternoon, creating a yard of space in the right-hand side of the box and seeing a deflected strike loop over a helpless Foderingham into the opposite side of the net.
On account of their maiden Premier League success over the Blades, Brighton rise above Newcastle United and West Ham United into seventh spot, while the basement-dwelling hosts remain rooted to the familiar position of 20th place.
Sheffield United now have a full week to rest up before next Sunday's trip to Wolverhampton Wanderers, which comes one day after Brighton welcome Everton to the Amex.