Michel Arteta's side could have drawn level on points with Tottenham with a victory after their rivals lost 1-0 to Brighton earlier on Saturday, but the result means they remain just outside the Champions League spots in fifth place.
It was a nervy start for Arsenal after Southampton's young striker Armando Broja weaved through the Arsenal defence and fired a ball across the face of goal in the opening minutes.
It was not long until the visitors called Souhampton goalkeeper Fraser Forster into action when Gabriel Martinelli slipped a low ball to Bukayo Saka, whose shot was superbly saved by the 34-year-old's hand.
However, it was Southampton who finally got the breakthrough in the 44th minute when Bednarek jumped onto a cutback from Mohamed Elyounoussi and slotted the ball past Aaron Ramsdale and into the net.
Arsenal continued to struggle after the break, lacking the connections and clinical touches in front of goal.