Cazorla held his nerve after Jose Fonte was penalised for tangling with Olivier Giroud in the box as Arsenal took all three points in a match they had appeared destined to draw.
Arsenal to sign Perez and Mustafi for over £50 million
Southampton took the lead in the 18th minute when Cech touched Dusan Tadic's shot onto the crossbar, only for the ball to ricochet down and bounce over the line off the goalkeeper's head.
Arsenal, who handed debuts to new signings Shkodran Mustafi and Lucas Perez, heaped on the pressure and were rewarded in the 29th minute when Koscielny slammed home a fine overhead kick as Southampton failed to clear their lines from a corner.
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