Saints fans were left disappointed in their first top-flight home game in seven years as a professional performance from the Latics saw them claim a win
The duo struck in the second half to leave the hosts without a point from their first two games back in the Premier League.
With Jack Cork still unavailable through injury, Nigel Adkins called Rickie Lambert and Steven Davis into the starting line-up, after both began last week's defeat to Manchester City on the bench before coming on to score, with youngster James Ward-Prowse and Jason Puncheon making way.
Kone was given a first start for the Latics, while Shaun Maloney and James McArthur both kept their places in the XI after being brought off as a precaution against Chelsea. Jean Beausejour recovered from a hamstring injury that ruled him out of that defeat, but started on the bench.
Saints had the better of the possession early on, and nearly got a breakthrough just before 15 minutes had elapsed. Lallana created a yard for himself with a neat turn before smashing a shot towards goal, and was only denied by a brilliant save from Ali Al-Habsi who tipped the ball onto the bar.
The home side continued to press after going close as Jose Fonte picked out Lambert with a long ball, the big striker getting a header on target but it proving comfortable for the Latics keeper.
Again, Lallana was at the focus of the hosts' next attack. The skipper stood over a free-kick on the far side, which he whipped in at pace, the ball evaded everyone as it swerved inwards and Guly Do Prado just failed to make contact at the far post.
As the game approached half-time, and the rainfall thickened, chances became few and far between. Di Santo wasted a rare good position as Wigan counterattacked, his shot, that may have been a cross, going wide. Then, after some decent link-up between Lambert and Jay Rodriguez, Lallana saw his attempted chip collected by Al-Habsi.
Wigan came out firing after the break and got the opening goal within five minutes. Shaun Maloney threaded a neat ball past the hosts' back-four as he turned sharply on the edge of the area. Di Santo was then clear of the back-line and smashed a strike past Kelvin Davis.
Southampton looked to hit back instantly as Danny Fox tried a volley from the edge of the box, but Al-Habsi was equal to it. As the half went on, again both sides failed to show any cutting edge on the attack.
Minutes later, though, Lambert nearly grabbed an equaliser. The big striker turned before unleashing an effort from range, the ball looked destined for the bottom corner but Al-Habsi stretched to get a fingertip on it and push it wide.
Saints continued to look for an equaliser but their failure to release the ball early enough made it difficult to create any real chances. Martinez's men also sharpened up in midfield, giving the home side less room to manoeuvre, and managed to keep Lallana quiet.
With just over 10 minutes to go, though, Jos Hooiveld clipped the crossbar with a looping header sent across the goal. The defender looked as if he thought he had levelled it, with Al-Habsi helplessly looking on.
Saints were then let off the hook as Wigan tried to pick them off on the counter. Kone raced clear of the defence and looked sure to make it 2-0, however he put his shot wide of the post when he was in the clear, though Fox may have done just enough to put the Ivorian striker off.
With under 10 minutes left Lallana tried to salvage a point for his side, hitting a low strike at goal after a decent run, Al-Habsi made the save, though couldn't collect the ball at the first attempt.
As Saints pushed forward in search of a late point, Wigan punished them on the counterattack as Kone got in behind the back-four after stealing the ball from Fonte, he then proceeded to smash the ball past Davis with ease, giving Martinez's men their first three points of the season, and leaving Adkins' on zero.