The France international scored both goals in a 2-0 victory at the Emirates Stadium, with Mauricio Pochettino quick to defend goalkeeper Artur Boruc after a costly error.
Giroud hassled Artur Boruc into making a comical error before slotting into an empty net and wrapped up the victory with a penalty late on.
His second goal moved the striker into double figures for the season in all compeitions, with Wenger quick to point out that the France international's appetite for a physical encounter is a key factor in his side's ability to to play attractive football.
"Olivier is a positive character who is always ready for a battle and he is maybe a little bit completely different to the rest of the team, but he gives us so much with his physical presence, with his link play which he has improved tremendously," he said. "He has the mental qualities. ... He has a positive attitude."
Wenger suggested prior to the win that the Gunners would not need to sign another striker to maintain their Premier League title challenge. And the Frenchman suggested a dearth of top-quality European-born forwards makes the progress of Giroud even more of a success story.
He added: "Europe produces still fantastic football players, but if you look well we don't produce in Europe any more strikers, or very few.
"All the big strikers today come from South America. You look at the summer period - you had Falcao going to Monaco [in a] big transfer, [Edinson] Cavani going to Paris Saint-Germain, [Luis] Suarez - everybody wanted to buy him. [Gonzalo] Higuain has gone to Naples [Napoli] for a lot of money. That's an area where in Europe it's very difficult to find because we produce less strikers.
"Even a country like Germany where I remember in every big club they had a big striker, they have produced fantastic players but they do not produce young strikers. Is that a consequence of the way we coach? Is that a consequence of modern life? I don't know."
Southampton boss Mauricio Pochettino, meanwhile, said he would support Boruc "100 percent" after his blunder allowed Arsenal to open the scoring.
"It's the way that we play," he said. "It's what I demand of him, I demand of all my players. That's the style that we play. There's nothing more that we can say. It's just a bad day, and it happens."