Saints' keeper Fraser Forster superbly saved from Alexis Sanchez before Szczesny kicked a loose ball while on the floor to Dusan Tadic, who poked in.
Southampton could have scored more, with Graziano Pelle's shot cleared from the line, as Arsenal stay fifth.
Saints' manager Ronald Koeman suggested his side could finish above Arsenal this week, a feat which seemed unlikely in the summer as almost £100m of players left.
A run of four league defeats from late November issued a reality check for the south coast club but dreams of a top-four finish once again seem realistic with 10 points taken from the last 12 on offer.
Saints in rare territory |
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Southampton started a calendar year in the top four in the Premier League for the first time since 1990. |
By contrast, Szczesny made errors which proved the difference between the teams.
First, the Poland keeper raced from his goal to confront Mane, but the Senegal striker - soon heading for the African Cup of Nations - curled in his third goal in as many games with a measured finish from the byline.
It was the seventh goal Arsenal have conceded from an error this season - only Everton, with 11, have more.
The Gunners nearly added to that tally when Szczesny was sold short by a backpass and Mathieu Debuchy was left to clear Pelle's prod off the line.
The Italy striker had more shots than any other Premier League player in the first half of the season and came close to adding to his eight league goals when he hit the post twice.
Arsenal, who suffered a first defeat in five league outings, had a strong case to be playing 10-men with a quarter remaining when Florin Gardos felled Sanchez as he raced in on goal.
Referee Craig Pawson issued only a yellow and although Arsenal introduced Theo Walcott for the first time since November, they barely threatened to reduce the arrears.
Graziano Pelle had five attempts at goal, hitting the post twice and seeing one cleared off the line
Per Mertesacker lunged but could not keep out Sadio Mane's curled effort as Sczesny raced back
Szczesny's second error landed at the feet of Dusan Tadic who scored his second league goal of the season
Arsene Wenger cut a frustrated figure on the touchline as his side struggled to hurt their hosts
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