United kicked off six points behind City after their rivals' victory over Tottenham Hotspur earlier in the day, and the champions were set to lose ground in the title race when Robin van Persie's second-half goal cancelled out Antonio Valencia's opener at the Emirates Stadium.
But United have a well-earned reputation for late heroics and they came up with yet another last-gasp winner thanks to a clinical finish from Welbeck with just nine minutes remaining. Welbeck's decisive intervention moved United back within three points of City and underlined their determination to stop Roberto Mancini's men wrestling the title away from their grasp.
In contrast, this was a dispiriting result for Arsenal, who pushed United hard in the second half, but have now lost three successive league games for the first time in five years and remain five points behind fourth-placed Chelsea.
United manager Sir Alex Ferguson was delighted at his team's attacking display. "We've had to play well today and for most of the game we did play well," Ferguson said. "Winning after City had won their game, that was the important thing. But I think we've done it in the right way - we were adventurous, positive and had belief in ourselves."
When news of City's win filtered through less than an hour before kick-off, Ferguson's players knew they were left with no margin for error. On the face of it, United had no reason to fear an Arsenal team thrashed 8-2 at Old Trafford in August and without a host of key players including Thierry Henry, who failed a fitness test on a calf injury.
Valencia heads home
This was a much closer encounter than that Manchester mauling, but United's ruthless streak ensured the result was the same. Phil Jones could only make it through the first 15 minutes before the United right-back was stretchered off after falling awkwardly while trying to close down Theo Walcott.
But England winger Walcott completely miscued Arsenal's only decent chance of the first half and Arsene Wenger's men were prone to lapses in defence, especially on the flanks where makeshift full-backs Johan Djourou and Thomas Vermaelen struggled.
Welbeck was the first to threaten when he got behind Arsenal's back four, only to lash his shot over with just Wojciech Szczesny to beat. United were in again when Patrice Evra twisted away from Alex Song and teed up Nani for a low strike that Szczesny saved at his near post.
Nani took advantage of more sloppy defending from Djourou to break into the penalty area before clipping a shot well wide of the far post. United's pressure finally told in first-half stoppage time when Nani fed Ryan Giggs down the left and the Welsh midfielder floated a teasing cross to the far post, where Valencia got above Vermaelen to power his header past Szczesny.
Wenger put Djourou out of his misery at half-time, sending on teenage defender Nico Yennaris in his place, and suddenly it was United's turn to show some defensive frailties instead.
Chris Smalling was under no pressure when he received the ball in his own half, but he slipped, allowing Tomas Rosicky to race away and pick out Van Persie. Van Persie has converted far harder chances during his magnificent run over the last year but he could only curl wide with the goal at his mercy.
Welbeck wins it
Aaron Ramsey and Alex Oxlade-Chamberlain went close as Arsenal found a head of steam at last. Welbeck's pace took him past Per Mertesacker before flicking a goalbound shot that the Arsenal defender hooked off the line.
And Arsenal took advantage of that escape as they swept the length of the pitch to equalise in the 71st minute. Laurent Koscielny started the move with a pass to Rosicky from deep inside his own penalty area.
Rosicky chipped the ball to Oxlade-Chamberlain on the left and he slipped an astute reverse pass to Van Persie, who guided a sublime low finish past Anders Lindegaard. Wenger replaced Oxlade-Chamberlain with Andrey Arshavin soon after, provoking a torrent of abuse from Arsenal fans unhappy with the Russian's form.
Unfortunately for Wenger, the criticism was spot on as Arshavin was at fault twice for United's 81st-minute winner. He allowed Valencia to ease past him into the penalty area and then could not make a challenge before the Ecuador winger laid the ball back for Welbeck to fire past Szczesny.
Wenger admitted his decision to replace Oxlade-Chamberlain with Arshavin was costly. "When it was 1-1 it looked like we would win but in the end one tactical mistake lost us the game," he said. "I do not want to dwell too much on that now but one second of inattention can cost you."