Manchester United's soon-to-be ex interim manager Ralf Rangnick apologised after a display he described as "humiliating" on the south coast.
Defeat left United trailing fourth-placed Arsenal by five points. With Rangnick's team having just one match left, they cannot finish higher than their current sixth place in the standings and all the teams above them have games in hand.
In fact, they could finish lower in the standings if they are leapfrogged by West Ham United, who are six points behind them but have two games in hand.
Moises Caicedo, Marc Cucurella, Pascal Gross and Leandro Trossard were all on target for Brighton as United suffered a fifth successive away defeat in the league to end any remaining hope of a top-four finish.
"It was a terrible performance. From the first until the last minute it was not enough. We can only apologise for this performance and a humiliating defeat," said Rangnick, who will hand over to Erik ten Hag at the end of the season.
United's 3-0 win over Brentford earlier this week had raised hopes that they might manage to sneak into Europe's premier club competition next season but it proved to be a short-lived pipe dream.
"[Monday's win over Brentford] was a different game, we just gave too much time and space (to Brighton)," added German Rangnick.
"We were never in a position where we could stop them playing through our lines. We told the players be as compact as they could but we couldn't stop it."
While defeats against top-four opposition Liverpool and Arsenal were not that surprising, the rout they suffered at Brighton left United's fans so angry that they chanted "you're not fit to wear the shirt" during Saturday's match.
"What we did today, what I did today, was not enough to be in a Manchester United shirt and I accept it," midfielder Bruno Fernandes told Sky Sports.
His manager added: "In games like today and at Everton [when we were beaten 1-0, there was a lack of energy and aggression. In the first three or four months we were on the right pathway, but in the last four weeks we have been really poor."