Kobbie Mainoo struck late to seal the victory after Liverpool fought back to erase United's early goals from Matheus Cunha and Benjamin Sesko.
Mainoo, whose career has seen new life under Carrick after he fell out of favour with previous manager Ruben Amorim, heaped praise on the interim boss.
"All the confidence he gives all the players, you want to follow him and fight for him and die for him on the pitch," Mainoo said.
Carrick has guided United to third in the table, a remarkable run that included victories over Arsenal, Manchester City, Chelsea and now Liverpool.
"It's been a good run, we've obviously beat some very, very good teams, and it's been challenging," Carrick said. "Listen, I love doing what I'm doing, it's a great position for me to be, and it feels pretty natural, if I'm totally honest, without being blasé, because it's a difficult role.
"But I kind of understand what it brings, and to be sat in this position is a good position to be."
Carrick stressed his focus had been on trying to get the best out of players, "and giving them the opportunity to go and perform."
"And then there's obviously the part of it today, seeing this place (Old Trafford) alive and bouncing at the end is a real joy and a pleasure to be in this position," he added.
Carrick praised the spirit of his squad.
"The pleasing thing to see that camaraderie and looking after each other, it was tested at 2-2, and seeing where they're really sticking in and fighting for each other, that's what we want," he said.
"We want a team to be proud of, and the boys have certainly gave us that."
Asked if he will be disappointed if he is not given the permanent job next season, the caretaker manager said: "We'll have to wait and see, that's all we can do.
"I think when we came in, Champions League, to be honest, was a little bit in the distance, and we want to try and get back into Europe," he said. "So, to be where we are with three games to spare is a good achievement."
But Carrick stressed that nobody was resting on their laurels.
"(Champions League) is a huge step, for so many reasons ... and there's a lot of satisfaction that comes with that," he said. "But that can't be everything. We want more than that, we want to expect more of that and consistently.
"And that's kind of the message really of: let's try and push on and keep improving. It's not just to sit back and think, 'that was all right. That was good. We've achieved something'."