United produced another strong performance at Old Trafford on Sunday, recording their seventh win in nine matches under Carrick and moving three points clear of Villa in the Premier League table.
The hosts struck first when Casemiro rose highest to head home from a corner delivered by Bruno Fernandes early in the second half. Villa responded through Ross Barkley, who marked his first league start in over a year with a well-taken equaliser.
But United remained the more dangerous side throughout the contest and regained the lead when Matheus Cunha finished calmly after a precise pass from Fernandes. Late in the game, Benjamin Sesko reacted quickest to a loose ball in the penalty area to seal the victory.
Fernandes' two assists took his tally to 16 in the Premier League this season, a new club record for United.
Speaking after the match, Carrick praised the performance and the team's response following their previous defeat.
"I was pretty happy with a lot of things. We looked a good team," Carrick told Match of the Day.
"Credit to them, they had a couple of spells in the game but for the most part we were dangerous. It was a big game coming off the defeat and the boys reacted incredibly well."
Carrick also highlighted the growing understanding between Fernandes and Casemiro, who combined again for United's opening goal.
"They have a really good understanding between them," he said. "Some things you can coach, some things you can't."
"We're delighted to have Bruno. He's proven over a period of time how impactful he is in the big moments. He's always there, he puts himself forward in training and games, and you can count on him."
"He's been through ups and downs here but it's never knocked him. He was there today to make the difference."
The win left United third in the Premier League, three points ahead of Villa and six clear of Chelsea in the race for European qualification.
Despite the strong position, Carrick insisted his players must remain focused with plenty of the season still to play.
"We can enjoy winning," he said. "It's a good feeling and something we want to keep getting used to. But we understand where we are and what it takes to win games like today."
"We are in a good position at the moment, but there is still a lot to play for."
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