Carrick and his players were booed off at the final whistle following a miserable 3-2 defeat featuring a blown two-goal lead.
Shea Lacey and Mason Mount put United two up by the 10th minute.
But the hosts suffered a woeful collapse as Charalampos Kostoulas reduced the deficit and Pascal Gross equalised before Maxim De Cuyper grabbed the winner.
For the first time in 50 years, United had lost a home game after leading 2-0.
It was another brutal blow for United after their controversial 1-0 home defeat against Manchester City on Sunday.
Carrick branded the third round loss unacceptable, but took the majority of the responsibility himself.
“We had the game exactly where we wanted it and let it get away from us. That’s the story of the game,” he said.
“There are a number of things we will look at and dissect. I know quite a bit about where it went wrong.
“We can’t accept that as a group. Listen, I take responsibility for that, with the performance towards the end of the second half. That is on me.”
Having led United to third place last season in his spell as interim boss, Carrick was expected to oversee a title challenge this term after being appointed on a permanent basis.
But United have already lost three times in six games in all competitions, putting Carrick under pressure to turn the tide at Fulham in the top-flight on Sunday.
Amid criticism of United’s close-season transfer recruitment, Carrick’s decision to make eight changes against Brighton exposed the lack of depth in his squad.
A defeat in west London at the weekend would leave United in a full blown crisis barely a month into the new campaign.
“We will all be better. We have to be better. We have a big game on Sunday that we have to prepare for. We are out of this competition. We’ve got to get back to winning football matches,” Carrick said.
“It doesn’t feel good at the moment, let me tell you. It doesn’t feel good. We can get back to winning football matches and we can do a lot of things better.
“We want to win and we want to win quickly. It’s as simple as that. The next game is Sunday and that is purely all I’m thinking about. We will get better with a number of things.”
Carrick was a decorated member of the United sides that won Premier League titles and the Champions League under Alex Ferguson.
His successful interim spell had raised hopes that he could be the man to lead United to their first English title since Ferguson retired in 2013.
But the former England midfielder, whose only previous managerial experience came with second tier Middlesbrough, is in danger of being dragged into a fight to save his job unless results improve quickly.
United co-owner Jim Ratcliffe, who runs the club’s football operations, has already sacked Erik ten Hag and Ruben Amorim during his reign.
Asked if he understood why the fans were so frustrated by United’s latest failure, Carrick said: “It’s pretty obvious isn’t it? I totally understand the feelings of the supporters.
“You have to accept it. I totally accept it. It is on me ultimately you know, as a team, what we look like, how we play, what results we get.
“I will take that. I will take the praise and I will stand here and take the criticism and the upset. I totally understand it tonight for the way the game went. We have got to win games.” (AFP)
