Matheus Cunha opened the scoring after six minutes, his left-footed strike deflecting off Alexis Mac Allister and beyond third-choice Liverpool goalkeeper Freddie Woodman. Benjamin Sesko doubled the lead on 14 minutes after Bruno Fernandes' header was pushed into his path.
Liverpool fought back after the break through Dominik Szoboszlai — benefiting from a calamitous error by substitute Amad Diallo — and Cody Gakpo, who profited from a poor kick by goalkeeper Senne Lammens to level the score. Kobbie Mainoo then settled it with a first-time finish from the edge of the box in the 77th minute, completing a memorable individual week in which he had also signed a new contract until 2031.
The result put Manchester United six points clear of Liverpool in the table and completed a league double over the Merseysiders for the first time since 2016. Carrick was asked whether sealing Champions League qualification with three games remaining was a source of pride.
"In some ways, yes. And in some ways I think the work that's been done, and the players, the way they've performed, and the teams that we've played against and got good results against — I think it's something that we can take a lot from, for sure. It was well in the distance, a little bit, if we're honest, at one point. So there has been a lot of good work to put us in this position."
He was careful not to allow the achievement to become a destination in itself.
"It's not something we've spoken about as a group, really. It's not something we over-celebrate. It's a step. We want more. We want to be finishing higher up the league."
Carrick has now won ten Premier League games since taking charge in January, with notable victories over Arsenal, Manchester City and now Liverpool. When asked how he has achieved such a turnaround, his answer was simple.
"Good players help. Good coaches, good staff. The connection that we've all had together has been pretty special. We've had to do it in different ways. We've won games in different ways, with different challenges, and today was a different one again. You just keep trying to do the right things."
On the tactical approach and the team's underlying identity since his arrival, Carrick offered a precise description of his priorities.
"We've scored a lot of goals — in pretty much every game. So a big focus of ours has been, if anything, on the out-of-possession work and trying to be harder to play against over a period of time. We knew they played a lot of midfielders today, tried to take the ball and control it. That gave us a big challenge. I thought we started the game brilliantly."
The two Liverpool goals — created by errors from Amad and Lammens — were described by Carrick as part of the game, without deflecting from the bigger picture.
"Mistakes happen. They're two fantastic players who have given us a lot to put us in this position. Fortunately for everyone, we managed to come out on top today. There was an underlying feeling of belief and it's tested in those moments. If someone makes a mistake, we'll help them out. Ultimately, it's quality football. The tactics is one side of it but it's actually the mentality, and the attitude towards it, that gets you through difficult and challenging times."
He reserved his final thoughts for the manner of the finish.
"For that period, it was of course challenging, but I thought we finished so strongly. To do it with games to spare — that's one thing — but we want to keep kicking on and finish the season strongly."
Carrick was also asked about Sir Alex Ferguson, who had been taken to hospital before kick-off after feeling unwell at Old Trafford. He said he had not received an update on the former manager's condition at the time of his post-match interview.
United sit third in the Premier League and face Brighton and Hove Albion, Aston Villa and Manchester City in their final three games of the season.