Ollie Watkins’ winner early in the second half lifted Villa up to seventh, just three points off the Premier League’s top four.
By contrast, 16th-placed Everton are looking over their shoulders, just one point above the relegation zone.
A run of no wins and just one goal in his last five games in charge cost Sean Dyche his job as Everton boss.
Moyes made his name during an 11-year stint in charge at Goodison Park from 2002 to 2013, a period that included a fourth-place Premier League finish in 2005 and an FA Cup final appearance in 2009.
However, he was given a rude awakening to the challenge that lies ahead of him in just maintaining the club’s 71-year stay in the top flight ahead of a move to a new 53,000 capacity stadium next season.
“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.”
Moyes raised a hand of recognition to all four corners of the ground as he was paraded before kick-off.
But there was no new manager bounce as Villa took control inside the opening minutes.
Morgan Rogers’ low effort was well saved by Jordan Pickford before James Tarkowski had to make a last-ditch tackle to deny Youri Tielemans.
Watkins wasted a glorious chance to score his first non-penalty goal since November when he latched onto Ashley Young’s short passback.
However, the England striker slotted wide with just international team-mate Pickford to beat.
Watkins strike
Everton’s seven Premier League goals from open play this season is by a distance the lowest in the division and their lack of both creativity and a clinical finisher was again in evidence.
Moyes has put his faith in Dominic Calvert-Lewin to score the goals that will keep his side up.
Yet the man who once rivalled Watkins to be England’s back up to Harry Kane has still not scored since these sides last met in September.
Calvert-Lewin came closest for the home side when his effort in first-half stoppage time was cleared off the line by Boubacar Kamara.
Watkins’ status as Villa’s talisman has come under threat by the form of Jhon Duran, while new signing Donyell Malen watched on from the stands.
But he made no mistake when handed a similar opportunity on 51 minutes to the one he wasted in the first half.