Southampton struck first, when a mistake from Shkodran Mustafi in the 17th minute allowed Shane Long to find the incoming cross and put it past Petr Cech.
Southampton were the better side early in the second half and got a deserved equaliser as Charlie Austin came off the bench to make it 2-2 in the 73rd minute following great work from Cedric.
But even without the influential Mesut Ozil and Jack Wilshere in their starting line-up they had the mental strength to take all three points against determined opposition.
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.
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But Southampton couldn't hold on, Welbeck's second of the day - a header from close range - securing all three points for the Gunners. Long is not the most prolific but he showed a poacher's instinct in opening the scoring with his second goal of the Premier League campaign, and his willingness to run the channels was vital against opponents who dominated possession.
And at the end of the first 45 minutes it was the Gunners that walked into the tunnel with a 2 - 1 lead. It was the Ireland striker's second league goal of the season and his first since January 2.
Welbeck stated afterwards: "It's a competitive game, Saints are in a hard position, same as us".
"I thought there'd be another opportunity to get a goal and get the victor".
The away side was denied another good chance to score when Mohamed Elneny headed clear under the crossbar from a Wesley Hoedt header before a super save from Cech to tip away a Long header.
We have gone through a hard spell, but because the players are proud we came back.