Both sides played out 2-2 draws just before the international break, with the Saints rescuing a point against Newcastle United while West Ham were pegged back by Crystal Palace.
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The Blues have picked up maximum points from their last four league matches to move into fourth position in the table, while Southampton have dropped to 13th following six straight losses in England’s top flight.
The much-changed Gunners, who have won the competition a record 14 times, fell behind midway through the first half and could not find a way back.
The 52-year-old was appointed at St Mary’s in December 2018 and has proved a popular appointment, turning around his side’s fortunes after a 9-0 home defeat to Leicester in October.