United knew they needed a win to keep close to leaders City and they looked on course for victory when Antonio Valencia nodded home Ryan Giggs' cross.
Arsenal rallied in the second half and pulled one back through Robin van Persie, but their hopes of a point were dashed nine minutes from the end when Welbeck steered Valencia's cross home from close range.
The game was not a patch on City's 3-2 win over Tottenham in terms of entertainment and quality, but Sir Alex Ferguson will be happy to have ground out another win which further proves he still has the know-how to formulate a title-challenging team at the grand old age of 70.
In reality, though, Arsenal made their task much easier thanks to an abject first-half defensive display from the Gunners' makeshift back four.
Despite the thrilling nature of the encounter between Tottenham and City - the two newest entrants in to the title race - earlier, Arsenal v Manchester United has proved to be a humdinger of a game in Premier League history.
Pizza-throwing tunnel clashes, touchline spats and on-field rucks have taken place in recent encounters and Sunday's match had even more added spice as the Gunners were out for revenge after their 8-2 humiliation at Old Trafford.
Both managers had predicted a cagey encounter on this occasion, however, and the first 10 minutes suggested they were right to do so.
Jonny Evans cleared an Alex Oxlade-Chamberlain cross and Anders Lindegaard easily gathered a shot from Aaron Ramsey.
Theo Walcott embarrassed Phil Jones with his quick feet, causing the defender to trip up and hit the deck and he had to come off for Rafael after just 15 minutes.
Per Mertesacker and Johan Djourou cleared dangerous United crosses and Valencia just failed to tap home a flick-on.
Giggs tested Wojciech Szczesny at his near post from a tight angle.
Oxlade-Chamberlain bamboozled Patrice Evra to sneak in to the United box and laid off to Walcott, but he blazed over from 18 yards.
Evra then easily beat Alex Song down the left and crossed for Nani but Szczesny saved.
Giggs played Wayne Rooney into the box and the striker went down after appearing to have his arm pulled by Song but Mike Dean waved play on.
The ability of United's wingers to skip past makeshift full-backs Vermaelen and Djourou was a particular source of frustration for the home support.
Nani easily beat the Swiss and whistled a sharp ball across the box.
In the opening minute of first half injury time Giggs was given space down the left flank and he floated a cross over to the back post where Valencia beat Vermaelen in the air to nod home.
Arsenal went off at half-time to a chorus of boos and Wenger, clearly unimpressed by Djourou's contribution, replaced the defender with 18-year-old Nico Yennaris who was making his league debut.
Van Persie was presented with an easy chance five minutes after the break when Tomas Rosicky pinched the ball off Chris Smalling and squared for him but he fired wide from 10 yards with the goal gaping.
Van Persie then tricked his way past three United defenders and squared for Rosicky but his drive was blocked by Evra.
Laurent Koscielny burst in to the United half and laid the ball off to Oxlade-Chamberlain but his drive flew barely an inch wide.
Welbeck raced through after a headed clearance from the United back line and slipped the ball past the advancing Szczesny only to see his goal-bound effort cleared off the line by Per Mertesacker.
The striker fell in the challenge from Arsenal's goalkeeper, but despite United's protestations, Dean did not pull play back for a penalty.
Then with 20 minutes to go, Arsenal stunned the visitors with an equaliser.
The Gunners broke through Oxlade-Chamberlain who squared for Van Persie and his reverse shot flew through Evans' legs and in off the post.
Arsenal fans reacted with fury when Wenger brought Oxlade-Chamberlain off two minutes later for Andrey Arshavin.
Nani hobbled off and was replaced by Paul Scholes after being injured in a tackle and Ferguson sacrificed substitute Rafael for Park Ji-sung.
The Emirates Stadium was shocked in to near silence with nine minutes left after Welbeck put the Red Devils ahead.
Valencia lost his marker Arshavin and played a clever one-two with Park before squaring to Welbeck, who bundled home from close range.
Van Persie flung himself towards a header in injury time but United cleared and Evans calmly guided Mertesacker's header in to Lindegaard's arms soon after as the visitors clung on for a vital win.