Santi Cazorla scored from the spot in added time to dispatch Southampton at the Emirates.
A Dusan Tadic free kick went in off Petr Cech’s back giving Southampton a 1-0 lead on 18 minutes.
But Laurent Koscielny levelled from an acrobatic overhead effort 15 minutes from halftime.
And Cazorla netted the winner from the spot after the referee adjudicated that Olivier Giroud had been brought down in the box.
Southampton is still seeking their first Premier League win of the season.
Arsenal: Cech, Koscielny, Perez (Giroud 62), Ozil, Walcott (Iwobi 75), Oxlade-Chamberlain (Sanchez 62), Monreal, Carzorla, Mustafi, Bellerin, Coquelin.
Unused subs: Ospina, Holding, Xhaka, Elneny.
Own goals: Cech (18), Carzorla (90+4)
Southampton: Forster, Bertrand, Fonte, Van Dijk, Cedric, Romeu, Clasie (Hojbjerg 64), David, Tadic (Ward-Prowse 80), Redmond, Rodriguez (Long 46).
Unused subs: McCarthy, Targett, Yoshida, Austin.
Bookings: Forster, Hojbjerg, Fonte, Van Dijk, Bertrand.