It had been a frustrating night for Sir Alex Ferguson's young troops until 20-year-old Welbeck, raised just a couple of miles away in Longsight, rose to guide home Tom Cleverley's cross in the 61st minute.
Welbeck's impudent back-heel set up Anderson for the second in the 76th minute and Tottenham had given up hope by the time Rooney netted from close range with three minutes remaining.
It leaves United trailing City by the narrowest of goal difference margins at the top of the Premier League.
With Ferguson sticking to his pledge of naming Phil Jones and Jonny Evans as the replacements for injured duo Nemanja Vidic and Rio Ferdinand, United's major surprise came on the bench, where Javier Hernandez was included two weeks earlier than expected after a bout of concussion.
Without Luka Modric, Tottenham began with a hesitancy which United came close to exposing through Cleverley.
However, as the half wore on, United's attacks reduced in potency, their approach play lacked direction and Spurs grew in confidence.
Rafael van der Vaart and Niko Kranjcar started to get hold of the ball in areas of some concern so, after a difficult start to his time in England, David de Gea picked the perfect time to show off the ability that persuaded Ferguson to pay £18million to sign him from Atletico Madrid.
There was a kamikaze element to his first intervention, as he tried to skip past Van der Vaart inside his own area.
In giving the home fans a heart scare, the 20-year-old was also showing off footwork of which the watching Edwin van der Sar would be proud.
De Gea made decent stops from Van der Vaart look routine rather than spectacular and with Tottenham opting against peppering him with high balls, as was the case at West Brom last week, De Gea was able to perform with an assurity so obvious in his opposite number Brad Friedel.
Now 40, the American was handed a contract by Harry Redknapp in the summer and was promptly allowed to extend his record for consecutive Premier League appearances to 276 matches.
Either side of half-time, Friedel denied former Aston Villa team-mate Ashley Young with smart stops as United, who dropped just two points on home soil throughout last season, struggled to impose themselves.
Part of the problem was that the most experienced member of their back four, Patrice Evra, has never found Aaron Lennon an easy opponent to subdue.
Unfortunately for Spurs, Lennon did not have a great evening.
After skipping clear of a stumbling Evra, Lennon could have rolled a pass to an unmarked Van der Vaart, who was screaming on his inside.
Instead, he attempted to drill a cross through a crowded penalty area to Gareth Bale and succeeded only in blasting straight at Evans.
It was one of those times when a manager must feel like screaming. Another came moments later when Van der Vaart went for goal from 35 yards without ever looking like getting anywhere near it.
After the frustration came that familiar sinking feeling for Spurs.
Appearing to run down a blind alley, Cleverley rescued the situation by whipping over a superb cross for Welbeck, who rose majestically between Michael Dawson and Younes Kaboul to glance a fine header into the corner.
Tottenham had Friedel to thank for not being blown away.
Twice the veteran denied Welbeck, the second occasion following a Nani shot he had also saved, and then Evans, who had launched a fizzing volley from his far post station.
United were worthy of their second though, which came thanks to a brilliant piece of skill from Welbeck, who returned Anderson's pass with an impudent back heel that the Brazilian coolly finished into the bottom corner.
It wasn't quite plain sailing as De Gea's struggles under the high ball returned to present Jermain Defoe with a chance that he smacked against the post.
But with Friedel saving from Nani and Rooney scooping over from three yards, United remained on top. Rooney's late header merely emphasised their superiority.
The white Red Devils t-shirt shows your passion for the red side of Manchester.
The 2012/13 Manchester United away replica soccer jersey.
Show your love for the Red Devils with a team poster in your room.
The Manchester United Supporters Soccer ball from Nike is great for any fan of the Red Devils.
The Manchester United Nike Swoosh t-shirt is too cool. Get in the Gear!
Check out the Manchester United Full Zip Jacket.
The Manchester United backpack is perfect to carry your gear to the field or take it to work or school.
Manchester United started out called Newton Heath with the green and gold colors. Go Retro!
Is a match jersey is not your style, check out the Manchester United vintage polo. Great for the pub or golf outing.
The 2012/13 Manchester United home replica soccer jersey.