Arsenal opened the scoring when Welbeck put Mesut Ozil through on goal and the German slotted home calmly.
Ozil repaid the favor by crossing for Welbeck to sidefoot in from close-range as the England striker scored for the first time in 16 club games.
Villa defender Aly Cissokho turned a Kieran Gibbs shot into his own net and the Midlands side failed to recover.
Arsenal were fortunate to escape with only a 2-0 defeat from Borussia Dortmund in the Champions League in midweek, while they had drawn three Premier League games before the trip to the Midlands.
They ended that stutter in form with devastating goal blitz in which two of their more criticized stars in Ozil and Welbeck did the damage as the Gunners made it 16 games at Villa Park without defeat.
Villa had been this season's surprise package and a win would have moved them top of the Premier League, but they are now third with the Gunners moving up to fourth.
The defeat was the first in the league suffered by Villa this season as a combination of Arsenal's first-half performance and a virus that had swept through their camp proved too much for them to handle.
Villa boss Paul Lambert had lost defender Nathan Baker and midfielder Ashley Westwood to the illness but said that, despite other players "not feeling the best", his side would "go for it".
The home side were initially true to his word as Gunners keeper Wojciech Szczesny had to palm a Fabian Delph shot around a post and then save a Ciaran Clark header at point-blank range.
But when Arsenal click into gear they are difficult to stop, which a stunned Villa found to their cost as they conceded three goals in quick succession.
Ozil put the Gunners in front when he ran onto a beautifully-weighted Welbeck pass before sliding a left-foot shot past keeper Brad Guzan.
The World Cup winner went from goalscorer to provider when he crossed for Welbeck to score with a first-time side foot from six yards, before a Gibbs shot was turned into his own net by Cissokho.
The pressure was on Welbeck before the game after he had missed glorious chances against both Manchester City and Dortmund but he helped relieve it the same venue where he scored twice for former club Manchester United in a 3-0 win last season.
Wenger had been forced to defend Ozil after some low key performances from the World Cup winner so far this season and the playmaker repaid his manager's faith as Arsenal comfortably closed out the game after their goal burst.
Aston Villa manager Paul Lambert on virus sweeping through side: "I don't know what kind of virus it is but it is certainly doing its rounds at the training ground so we will clear that place for a couple of days.
"To play Arsenal you need everyone fit and we never had that, but all credit to them for the second half - the lads were running on empty.
"The players have been brilliant at the start of the season and if somebody had said we would have 10 points by this time I would have taken it."
Arsenal manager Arsene Wenger: "We had a good work-rate and concentration from the start. We had a very good first half when we were in control and dominated the game.
"They [Danny Welbeck and Mesut Ozil] combined very well, especially in the first half. Our plan today was to play Mesut behind their midfield and it worked.
"I'm confident we will go from strength to strength now."
Liverpool vs West Ham
Early goals from Winston Reid and Diafra Sakho proved too much for the visitors, who pulled one back through Raheem Sterling before the break.
Liverpool pressed hard after half-time.
However, Hammers keeper Adrian made a succession of saves and Morgan Amalfitano wrapped up a first home win for West Ham in the 88th minute.
Sam Allardyce's men had lost both previous home games this season and taken just four points from their last eight games against these opponents.
But they could not have started much better against Liverpool.
Reid's first goal since October 2013 was a close-range effort after James Tomkins had headed Stewart Downing's free-kick into the six-yard box.
West Ham's second was more speculative as Sakho floated the ball over Simon Mignolet from the right, the ball dropping into the corner before Enner Valencia could reach it.
A difficult start for the visitors also included a yellow card for Mario Balotelli, who had a head-to-head confrontation with Hammers goalkeeper Adrian, and the exit of the injured Javi Manquillo, who was replaced by Mamadou Sakho.
Liverpool were trying to shrug off the exertions of their successful midweek return to the Champions League.
And they did have hope through their remarkable scoring record away from Anfield.
When Sterling lashed home the rebound for his third goal of the campaign after Balotelli's shot had initially been blocked, it was their 34th goals in the last 12 away games and created a club record of scoring in 17 successive top-flight matches on the road.
Valencia forced Mignolet into a good save just after Dejan Lovren had been patched up to play on following a heavy collision with his own team-mate, Sakho.
The introduction of Adam Lallana at half-time for Lucas - and the consequent tactical switch - suggested Brendan Rodgers was unhappy with his team's efforts.
On his first Liverpool start for almost two years, Fabio Borini made little impact until a low shot at the start of the second half brought a low save from Adrian.
The Italian curled a shot narrowly over as Liverpool continued to press forward in search of an equaliser.
Then Balotelli, who has now scored just one goal in his last 19 Premier League appearances, was thwarted by Adrian.
Rickie Lambert was introduced for the last push forward.
However, Liverpool were vulnerable on the counter-attack and as time ticked down, Downing's pass released Amalfitano, who applied the clinical finish.
It was the first time the visitors had lost successive league games since last December.
West Ham manager Sam Allardyce:
"That performance today just shows what level and what capabilities we might be able to achieve this season.
"To score two goals in that early period really stunned the Liverpool players. We really took them to task. We played outstandingly well and really didn't let Liverpool play at all."