In the first match of Moyes’ second spell at Goodison Park, Ollie Watkins scored the only goal to continue the Toffees barren streak in front of goal.
A run of no wins and just one goal in his last five games in charge cost Sean Dyche his job as Everton boss.
Defeat leaves Everton, who have not dropped out of the top flight for 71 years, just one point and two places above the relegation zone.
“I’ve got no illusions of the job I’ve to do. We couldn’t score a goal tonight again,” said Moyes.
“We’ve got to show an awful lot more quality on the ball and play better with the ball.
“I’ve told the players that. We just didn’t play well enough and didn’t retain the ball as well as we should have done.”
Moyes had previously said he backed Dominic Calvert-Lewin to score the goals to get Everton out of trouble.
However, the former England international wasted a big chance to equalise in stoppage time as his wait for a goal since the sides last met in September goes on.
“I’ve said to Dom to see if he can get himself in goalscoring positions. I think he can show everybody that he’s a really hard-working centre-forward,” added Moyes.
“He had a pretty big miss at the end which was probably the big chance for him. Overall, we got our opportunities, they didn’t get many at all either. We just didn’t quite finish it.”
Victory lifts Villa up to seventh and within three points of the top four.
And Unai Emery was pleased to end a wait since October for an away win in the Premier League.
“This victory is very important. We have started the year winning at home, in the Premier League and in the FA Cup. Away, we were in a bad spell and had to break it,” said the Spaniard.
“To get three points here is very difficult and we’re very happy.”
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