Boos rang around Old Trafford at the final whistle as United fell to their third defeat of the season, and second in as many games following Sunday's loss to Manchester City.
"We had the game exactly where we wanted it and let it get away from us in a big way," United coach Michael Carrick told Sky Sports. "We can't accept that as a group. I take responsibility for it."
United led 2-0 after 10 minutes against Brighton through quickfire goals from Shea Lacey and Mason Mount.
But Brighton launched a stunning comeback with Charalampos Kostoulas pulling one back before halftime before Pascal Gross and Maxim De Cuyper secured victory after the break.
Defeat means United have suffered elimination from the competition at the earliest stage for the second season in a row.
United have also won two of six games in all competitions and only once in the Premier League.
After leading a resurgence last season that secured a return to the Champions League, Carrick has not been able to maintain that momentum and jeers from the home fans underlined growing unrest.
"It doesn't feel good at the moment, let me tell you," Carrick said. "But we can get back to winning football matches and we've got to do a lot of things better.
"I totally understand the feelings of the supporters. We have to accept it. I totally accept it. It's on me ultimately, as a team, what we look like, how we play and what results you get. I'll take that. You take the praise. I'll stand here and take the criticism and the upset."
At Coventry Building Society Arena, Tammy Abraham scored twice in his first start of the season as Aston Villa booked their place in the fourth round with a 3-1 victory over Coventry.
Abraham started over summer signing Nicolas Jackson and cashed in on his chance to pile more early season misery on to Frank Lampard's side who were beaten for a second time at home in the space of a week.
Both teams have shared early season struggles in front of goal in the Premier League, with Coventry still yet to find the net while Villa only scored once, so few would have been blamed for not expecting goals.
But Villa have saved their best goalscoring form for the cup matches so far this season and Abraham slotted Villa in front before Victor Torp's sublime free-kick got Coventry level.
Ross Barkley punished the hosts from a corner five minutes before the break to give the visitors the lead heading into the interval.
And set-pieces proved to be the Sky Blues' downfall as they conceded from a corner for a second time in the contest when Abraham tapped home from John McGinn's delivery.
Everton beat second-tier Wolves 1-0 at home to progress with Charly Alcaraz scoring in the 80th minute, while League Two Fleetwood beat Chris Wilder's Sheffield United 1-0.
Meanwhile, Liverpool will host Chelsea in a blockbusting all-Premier League tie in the fourth round.
DRAW: (All ties to be played week commencing October 26).
Liverpool vs Chelsea
Fleetwood vs Arsenal
Brighton vs Manchester City-Norwich
Everton vs Newcastle
Sunderland vs Brentford
Fulham vs Crystal Palace
Bournemouth vs Aston Villa
Bradford vs Peterborough
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