The Red Devils completed a Manchester double over North London to keep pace with rivals City at the top of the Premier League.
Chances were few and far between in the first period with Nani having United's best effort on goal.
The Portuguese winger cut inside in a crowded penalty area but his strike was saved at the near post by Arsenal keeper Wojciech Szczesny.
There was little to shout about for the hosts although Alex Oxlade-Chamberlain, who was making his first Premier League start, was lively down the Gunners left.
Just as it appeared the match would be entering the break goalless, United took advantage of sloppy Arsenal defending to go in front.
Arsenal equalised with 70 minutes gone courtesy of Robin Van Persie, who applied the finishing touch to a flowing counter attack.
In his own box, Laurent Koscielny dispossessed Rafael before finding Tomas Rosicky with a fine ball. The Czech Republic international spread the ball to Oxlade-Chamberlain, who slid in to the Dutchman to strike home.
United restored their advantage when Welbeck smashed home from close range after Valencia split the Arsenal defence with a mazy run.
German defender Per Mertesacker went close to levelling late on but Sir Alex Ferguson's side hung on to claim the three points.