United looked in complete control after Shea Lacey and Mason Mount scored inside the opening 10 minutes. However, their two-goal advantage quickly disappeared as Brighton produced a remarkable comeback.
Charalampos Kostoulas pulled one back before Pascal Gross levelled the score. Maxim De Cuyper then completed Brighton's turnaround by scoring the winning goal.
The result ended United's EFL Cup campaign in the third round and left the home supporters furious, with the players and Carrick booed at the final whistle.
United had not lost a home match after leading 2-0 for 50 years, adding to the disappointment of another poor result.
Carrick did not hide from the criticism and admitted his team allowed the match to slip away.
"We had the game exactly where we wanted it and let it get away from us. That's the story of the game," Carrick 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."
The defeat came only days after United lost 1-0 at home to Manchester City in the Premier League, increasing the pressure on Carrick just six games into the season.
The manager made eight changes for the Brighton match, but the result highlighted concerns about the strength of his squad beyond his first-choice players.
Carrick knows United must respond quickly, with a Premier League trip to Fulham next on Sunday.
"We will all be better. We have to be better. We have a big game on Sunday that we have to prepare for," he said.
"We are out of this competition. We've got to get back to winning football matches.
"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."
Carrick's successful spell as interim manager last season, when he guided United to third place, led to his permanent appointment and raised hopes of a title challenge.
Those expectations have quickly changed following three defeats in six matches in all competitions.
The former United midfielder also understood why the supporters made their feelings clear after another disappointing night at Old Trafford.
"It's pretty obvious isn't it? I totally understand the feelings of the supporters," Carrick said.
"You have to accept it. I totally accept it. It is on me ultimately, 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."
United now have little time to dwell on the EFL Cup exit. They travel to Craven Cottage on Sunday to face Fulham in the Premier League, with Carrick demanding an immediate response from his players.