With Dutch international Robin van Persie out for a month, Welbeck was picked to lead the line and helped United win for the first time in five Premier League games.
A fringe player under Moyes, following on from Alex Ferguson's last season when he managed just two goals, Welbeck produced two clinical finishes to double his tally to four in this campaign.
Welbeck, a regular scorer for England when picked by Roy Hodgson, has been questioned for his lack of consistency and killer instinct in front of goal following an impressive breakthrough 2011-12 season when the young striker formed a lethal partnership with Wayne Rooney.
In mitigation, Welbeck has often been forced to play out of position, wide on either flank, but after his goals on Sunday said he did not mind where he played.
"I'm happy to play my part anywhere in the team, be it left or right wing as well. I'm delighted with the goals today," he told Sky Sports.
"It was a good all-round team performance from the boys, played some good football and got the goals. We've set the platform to build on."
Moyes singled out Antonio Valencia's impressive performance against Villa but said Welbeck could be satisfied with his efforts.
"Danny Welbeck is someone who we know needs to continue to keep scoring goals," said the Scot.
"I thought he was good today. I thought he was a threat, he maybe could have got another one or two himself. It would have been nice for him to get a hat-trick... but two is a good tally for him."
Midfielder Tom Cleverley scored United's third goal early in the second half, exactly one year to the day since his last league goal in a red shirt.
"Obviously it plays on your mind... when your a midfielder you need to score a few goals. It was good finish... a year's worth of not scoring came out in one celebration," said the England international who admitted this week that he had not come up to his own high standards in recent games.
"I played quite well today, kept the ball, kept us ticking at a high tempo. I've had a couple of average performances this season but I want to kick from here now. Hopefully, this is the start of a good run."
Moyes has come in for his fair share of criticism in a tough baptism in the Old Trafford hotseat. Victory lifted United up to eighth place, 10 points behind leaders Arsenal, and the United manager said his side must keep improving.
"We know we have to play better than we did today," he said/ "It continually has to improve and it will improve and I'll make sure it gets better in the coming months.
"I think we'll play much better than that today, I've told the players that's the least we can do and we've got to keep going and keep winning more games."