United won 2-0 thanks to goals from Bryan Mbeumo and Patrick Dorgu in a dominant performance.
Carrick remarked afterwards: "I have to say it feels great. You could see how much our boys wanted it. It's great to feel that.
"I'm not really sure if they even had a shot on target... did they? A couple of blocks. I think we defended really well and we looked hugely dangerous on the break."
On the performance of skipper Bruno Fernandes in a No10 role, Carrick also stated: "I've known him a long time now and I thought he was fantastic today.
"The way he played the position... he's just got so much quality. Playing in that position, he can be a real threat."
Carrick admitted he insisted on rushing Harry Maguire back from injury ahead of schedule to start in a back four today.
He continued: "It was a big ask for H, I have to say, credit to him. People take it for granted sometimes what players do. For him to play today and come through it. It was a bit of a calculated gamble because he only trained for two or three days.
"It just shows you what it means for him. He was fantastic. Him and Licha were so solid at the back."
Looking at Mbeumo and his fellow attackers, the manager said: "We wanted to have a real attacking intent and I thought the front four, backed up by what was behind them, were pretty special today."
Mason Mount, as a late substitute, had a third goal ruled out for offside late on, much to Carrick's frustration.
He added: "It would have been lovely to finish, you know, on a high with a goal, but really, clean sheet was vital and the energy in the stadium and to be able to...
"I think for everyone to feel it, you know, and it means a lot and we've got to use that for fuel to excite us and drive us on to the next one."