In what was a welcome return to form for United, they claimed their first Premier League win since they beat Arsenal over a month ago.
After a bright start from Villa, it was United who went ahead on 15 minutes as Welbeck finished from close range after Adnan Januzaj's header had cannoned back off the post.
Just minutes later and Welbeck was on hand to convert his second as he slid home to meet Antonio Valencia's cross.
The scoring was completed not long into the second half, and it was another well-worked team goal as Tom Cleverley exchanged passes with Wayne Rooney before firing past Brad Guzan.
United were also boosted by the return of Scottish midfielder Darren Fletcher, who made his first league appearance in nearly 12 months when he came on in the second half.
David Moyes's side had been staring at a hat-trick of defeats for only the third time in the entire Premier League era.
Whilst presiding over a relegation battle in the season after replacing Sir Alex Ferguson is evidently not going to happen, for Moyes, he can only hope it is a victory that kickstarts the season.
Not that he will meet many opponents like Villa, for whom striker Christian Benteke was a major disappointment.
Valencia gave Antonio Luna a torrid time and, with the Spanish defender unsettled, Rafael was able to surge to the by-line on the quarter-hour mark and send over a superb cross that was met perfectly by Januzaj.
The teenager's header flashed past Guzan and, when it came bouncing back off a post, Welbeck was racing in to apply a first-time finish to the rebound.
Within three minutes, the visitors had struck again.
This time Valencia provided the opportunity, with Welbeck sliding between defenders to touch home.
It was the perfect way for United to answer their critics, and Welbeck to silence a few of his.
The England striker revelled in the responsibility of leading the attacking line as Robin van Persie starts his month on the sidelines, with Rooney dropping so deep at times, he was almost assuming the Paul Scholes creative role.
Rooney did skim the outside of a post after Valencia had opted to pass rather than shoot after driving deep into the area.
Welbeck spurned a glorious chance to claim his first senior hat-trick from a very similar position after Valencia had once again proved too hot to handle.
It was an all-too-familiar story for Villa, for whom Benteke was a shadow of the player who terrorised defences throughout last season.
A confrontation between David de Gea and Gabriel Agbonlahor, making his 300th Villa appearance, at the start of the second period suggested the hosts had received a blasting from manager Paul Lambert.
But pretty soon, Rooney had powered forward and slipped a pass to Cleverley, who drove home his first Premier League goal in almost exactly a year.
Soon afterwards, the England midfielder was throwing himself in front of an Andreas Weimann shot as United attempted to preserve their healthy advantage, a sign in itself of the determination in the away ranks to right the wrongs of a dreadful few weeks.
The return of Fletcher after almost 12 months out with, and virtually three years affected by, a chronic bowel condition, merely added to the party atmosphere amongst the away support.
De Gea denied Karim El Ahmadi after Welbeck had floated a chip onto the roof of Guzan's goal but, by then, thousands of Villa fans had already started to head for the exits.