More to come from United, says Moyes
Moyes had come in for plenty of criticism after the champions slumped to successive home defeats against Everton and Newcastle United.
Yet United avoided a third top-flight defeat in a row as Danny Welbeck scored twice and Tom Cleverley claimed his first league goal for a year to see off Villa.
The former Everton manager was encouraged by his side's display against Paul Lambert's men, but insisted United can still get better.
Moyes said: "We are getting closer to it, but it has to be more consistent and I'm still looking for that to improve again.
"We could have scored more goals, but I think the all-round performance was much more like that.
Moyes was pleased for Welbeck, who found the net in the Premier League for the first time since August.
He added: "I think for Danny it's really important, because he's a forward and he needs to get himself goals.
"It's always been labelled at him, but I thought today he got two poachers-type goals and could have had another one.
"I thought all-round he played well for the team, he was a threat going in behind and I was pleased for Danny, I thought he did a good job."
Darren Fletcher made his first appearance since Boxing Day when he came off the bench with just over 20 minutes remaining.
Fletcher got a great reception on his return after a lengthy spell on the sidelines due to a bowel problem, but Moyes is not expecting too much, too soon from the Scotland midfielder.
He said: "I think anybody who returns from an illness to play professional football for a club like Manchester United, it shows a lot about the boy himself.
"The time and hours he's been through where he's had to be on his own, work on his own in a hospital with operations, I think today was a big message to a lot of people - if you're not well then keep believing and you can get right.
"But more importantly he came on and didn't look out of place; he stroked the ball around and I don't know if he gave the ball away in the time he was on.
"I don't want to put pressure on and say it's Darren Fletcher back, because you just never know.
"We've been looking for more reserve games; we've got him three or four games and he seemed to cruise through the reserve-team games, so we thought what else can we do? We might need to get him back in the team and see how he does."