Goal-shy striker Shane Long had given the visitors a surprise early lead, but striker Pierre-Emerick Aubameyang and Welbeck earned the hosts a half-time lead and Southampton substitute Charlie Austin equalised to set up the dramatic finale.
Southampton are three points behind 17th placed Crystal Palace but with a game in hand. While Arsenal's last chance of silverware and their best chance of Champions League qualification hinges on their European campaign, they will not want their Premier League status to dissolve. I think he will miss these games for sure, but he could still play at the end of the season'.
A sparse home crowd bristled as their side laboured but Aubameyang hauled them back into the contest just before the half-hour mark, the finish on the stretch scruffy but the back-flick into his path from Welbeck inspired.
Welbeck headed in an 81st-minute victor to secure Arsenal a sixth successive win in all competitions, though Wenger admitted fatigue was to blame for his side's below-par display.
'He's got knee ligament damage, ' Wenger revealed after the game.
"It's a competitive game, Saints are in a hard position, same as us", he said.
"I've got the utmost respect for Arsene".
But the England worldwide made no mistake nine minutes from time when he headed Iwobi's delivery to the back post beyond McCarthy.
"I thought I would get another opportunity to get a victor and I kept focused on that".