Wenger had made seven changes with Thursday’s Europa League quarter-final second leg against CSKA Moscow in mind, and his side trailed when Shane Long capitalised on a first-half mix-up between keeper Petr Cech and centre-back Shkodran Mustafi.
But the hosts recovered with Pierre-Emerick Aubameyang prodding in his sixth goal since joining from Borussia Dortmund for £56m in January to level before Welbeck – who created the equaliser with a clever flick – put them ahead.
Southampton then drew level when substitute Charlie Austin turned in Cedric Soares’ cross but Welbeck headed Arsenal back in front after both sides missed a string of chances, leaving the visitors deep in relegation trouble.
“It’s not easy because we played against a team who fights not to go down, mentally it is very difficult,” Wenger said.
“Some of our players had heavy legs but we still won the game when we were questioned, I really like the spirit in this team.
“Every game you win makes you stronger, it’s important that you keep your run going. We have gone through a difficult spell, but because the players are proud we came back. Now we hope to keep this going.”
He singled out Welbeck’s contribution in the game at Emirates Stadium after the 27-year-old scored for the first time in the league since September, then got the winner after shooting high over the bar from close range four minutes earlier.
“I think I have repeated that I really like Danny Welbeck,” Wenger said.
“I am very pleased for him because he is a guy who went through a very difficult time and has shown exceptional mental strength to come back.”
Mark Hughes’ men, who had defender Jack Stephens sent off in added time before Arsenal’s Mohamed Elneny was also dismissed, remain 18th in the table and three points from safety.
The south-coast outfit also had a second Long goal ruled out for offside and defender Wesley Hoedt saw his header cleared off the line by Elneny.
Goalkeeper Alex McCarthy had kept the visitors’ second-half hopes alive with a string of impressive stops to deny Aubameyang, Granit Xhaka and Alex Iwobi.