Casemiro's opener in the second-half was canceled out by Villa's Ross Barkley before Matheus Cunha and Benjamin Sesko struck to earn the 3-1 win on Sunday.
Carrick remarked afterwards: "It feels good. We did a lot of things that we're really happy with. We knew it was a big game coming into it, to get back to winning ways and the boys responded really well (to the Newcastle United defeat)
"It was an important game, for sure. Going into the match, we've never been in that situation, with a defeat coming into the game, so it could go either way, and it makes such a difference really. But the boys have dealt with all of that really well, they have done all week. So I had a good feeling about today.
"We were in a really good place going into the game, focused but confident. It showed and I thought, as a team, we played really well today and got what we deserved."
Skipper Bruno Fernandes laid on two of the three goals for United and Carrick also said: "He's done it for quite some time and really put himself in a position where he can make the difference.
"The numbers are all good, and he gets the plaudits in terms of chasing assists but, for me, it's more about the squad and I don't mind whoever scores, as long as we're winning. It shows the impact he's had and he was important for us today.
"He's had so many big moments, to score a goal, to make a pass. Training or games, he's always there. The pass to Matheus was fantastic to cap off a good performance all round."