Shane Long gave them the lead after 17 minutes, nipping in ahead of defender Shkodran Mustafi.
Welbeck headed home the victor in the 81st minute to settle an eventful game at the Emirates, having previously made it 2-1 with a deflected shot in the first half.
Welbeck missed a great chance to put Arsenal back ahead when he fired over from three yards out after Wilshere had hooked the ball across goal.
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We did so in full compliance with our legal obligations to preserve messages", the company said. The tampering wasn't publicly announced; and the users weren't individually informed, either.
Arsenal did finish with 10 men when Mohamed Elneny saw red for a hand in the face of Cedric Soares, but Wenger believed the game should be remembered most for Welbeck's first Premier League goals since September a year ago.
A precious point was there to be snaffled and, having ruined their first league game under Mark Hughes with a wretched collapse against West Ham last Saturday, that unfamiliar sensation Southampton were experiencing was a surge of optimism.
Both Aubameyang and Granit Xhaka forced top end saves out of McCarthy in the first three minutes of the second half.
The last time the forward managed to hit the back of the net away from St Mary's Stadium was on 11 February 2017, during Southampton's 4-0 win over Sunderland. Long turned in Højbjerg's wayward shot, only for the flag to go up for offside.
- In fact, only Wayne Rooney (7) and Robin Van Persie (6) have scored more Premier League goals against Petr Cech than Shane Long (5).
Man City were offered Paul Pogba in January - Guardiola
But have United the ability to press and harry City the way Liverpool did? "These are the games you play the whole season for. United have lost eight of the last 13 league Manchester derbies , as many as in the previous 49 encounters.
Despite today's scoreline, Arsenal have struggled in the Premier League this year, with the club now occupying 6th spot in the table.
The game looked to be petering out before an on-field coming-together between Jack Wilshere and Jack Stephens led to the latter being sent off.
"Going into the Europa League on Thursday, it's going to be a hard tie even though we won the home game [4-1]".
"His game is about going past people, about taking them on, and overall I believe he's shown he has the necessary tactical and technical level".
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